Properties

Degree 48
Conductor $ 41^{24} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·2-s − 6·3-s + 44·4-s − 10·5-s + 60·6-s − 8·7-s − 110·8-s + 18·9-s + 100·10-s − 16·11-s − 264·12-s + 80·14-s + 60·15-s + 168·16-s + 8·17-s − 180·18-s + 16·19-s − 440·20-s + 48·21-s + 160·22-s + 12·23-s + 660·24-s + 31·25-s − 44·27-s − 352·28-s + 40·29-s − 600·30-s + ⋯
L(s)  = 1  − 7.07·2-s − 3.46·3-s + 22·4-s − 4.47·5-s + 24.4·6-s − 3.02·7-s − 38.8·8-s + 6·9-s + 31.6·10-s − 4.82·11-s − 76.2·12-s + 21.3·14-s + 15.4·15-s + 42·16-s + 1.94·17-s − 42.4·18-s + 3.67·19-s − 98.3·20-s + 10.4·21-s + 34.1·22-s + 2.50·23-s + 134.·24-s + 31/5·25-s − 8.46·27-s − 66.5·28-s + 7.42·29-s − 109.·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(41^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(41^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(48\)
\( N \)  =  \(41^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{41} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(48,\ 41^{24} ,\ ( \ : [1/2]^{24} ),\ 1 )$
$L(1)$  $\approx$  $6.85571e-6$
$L(\frac12)$  $\approx$  $6.85571e-6$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 41$,\(F_p(T)\) is a polynomial of degree 48. If $p = 41$, then $F_p(T)$ is a polynomial of degree at most 47.
$p$$F_p(T)$
bad41 \( 1 + 4 T - 48 T^{2} - 584 T^{3} - 1562 T^{4} + 25200 T^{5} + 246299 T^{6} + 113818 T^{7} - 6812644 T^{8} - 49201416 T^{9} - 35560466 T^{10} + 1161094538 T^{11} + 9242381481 T^{12} + 1161094538 p T^{13} - 35560466 p^{2} T^{14} - 49201416 p^{3} T^{15} - 6812644 p^{4} T^{16} + 113818 p^{5} T^{17} + 246299 p^{6} T^{18} + 25200 p^{7} T^{19} - 1562 p^{8} T^{20} - 584 p^{9} T^{21} - 48 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \)
good2 \( 1 + 5 p T + 7 p^{3} T^{2} + 115 p T^{3} + 3 p^{8} T^{4} + 275 p^{3} T^{5} + 5581 T^{6} + 6395 p T^{7} + 13413 p T^{8} + 12985 p^{2} T^{9} + 23341 p^{2} T^{10} + 39095 p^{2} T^{11} + 244623 T^{12} + 178945 p T^{13} + 61293 p^{3} T^{14} + 157685 p^{2} T^{15} + 383343 p T^{16} + 447285 p T^{17} + 1034701 T^{18} + 311915 p^{2} T^{19} + 410931 p^{2} T^{20} + 593585 p^{2} T^{21} + 1798923 p T^{22} + 2711865 p T^{23} + 7863425 T^{24} + 2711865 p^{2} T^{25} + 1798923 p^{3} T^{26} + 593585 p^{5} T^{27} + 410931 p^{6} T^{28} + 311915 p^{7} T^{29} + 1034701 p^{6} T^{30} + 447285 p^{8} T^{31} + 383343 p^{9} T^{32} + 157685 p^{11} T^{33} + 61293 p^{13} T^{34} + 178945 p^{12} T^{35} + 244623 p^{12} T^{36} + 39095 p^{15} T^{37} + 23341 p^{16} T^{38} + 12985 p^{17} T^{39} + 13413 p^{17} T^{40} + 6395 p^{18} T^{41} + 5581 p^{18} T^{42} + 275 p^{22} T^{43} + 3 p^{28} T^{44} + 115 p^{22} T^{45} + 7 p^{25} T^{46} + 5 p^{24} T^{47} + p^{24} T^{48} \)
3 \( 1 + 2 p T + 2 p^{2} T^{2} + 44 T^{3} + 77 T^{4} + 70 T^{5} + 2 T^{6} - 230 T^{7} - 710 T^{8} - 868 T^{9} - 28 p^{2} T^{10} + 268 p^{2} T^{11} + 10961 T^{12} + 2612 p^{2} T^{13} + 36392 T^{14} + 48602 T^{15} + 15776 T^{16} - 32548 p T^{17} - 266858 T^{18} - 569408 T^{19} - 713930 T^{20} + 82670 T^{21} + 1900202 T^{22} + 6019306 T^{23} + 13437106 T^{24} + 6019306 p T^{25} + 1900202 p^{2} T^{26} + 82670 p^{3} T^{27} - 713930 p^{4} T^{28} - 569408 p^{5} T^{29} - 266858 p^{6} T^{30} - 32548 p^{8} T^{31} + 15776 p^{8} T^{32} + 48602 p^{9} T^{33} + 36392 p^{10} T^{34} + 2612 p^{13} T^{35} + 10961 p^{12} T^{36} + 268 p^{15} T^{37} - 28 p^{16} T^{38} - 868 p^{15} T^{39} - 710 p^{16} T^{40} - 230 p^{17} T^{41} + 2 p^{18} T^{42} + 70 p^{19} T^{43} + 77 p^{20} T^{44} + 44 p^{21} T^{45} + 2 p^{24} T^{46} + 2 p^{24} T^{47} + p^{24} T^{48} \)
5 \( 1 + 2 p T + 69 T^{2} + 72 p T^{3} + 1543 T^{4} + 1132 p T^{5} + 18074 T^{6} + 10104 p T^{7} + 122763 T^{8} + 49416 p T^{9} + 350408 T^{10} - 1682 p T^{11} - 2446121 T^{12} - 2337392 p T^{13} - 39548786 T^{14} - 22209758 p T^{15} - 268607914 T^{16} - 111457552 p T^{17} - 933994062 T^{18} - 193966318 p T^{19} + 935721583 T^{20} + 1866705868 p T^{21} + 34987620397 T^{22} + 19948878894 p T^{23} + 240167281151 T^{24} + 19948878894 p^{2} T^{25} + 34987620397 p^{2} T^{26} + 1866705868 p^{4} T^{27} + 935721583 p^{4} T^{28} - 193966318 p^{6} T^{29} - 933994062 p^{6} T^{30} - 111457552 p^{8} T^{31} - 268607914 p^{8} T^{32} - 22209758 p^{10} T^{33} - 39548786 p^{10} T^{34} - 2337392 p^{12} T^{35} - 2446121 p^{12} T^{36} - 1682 p^{14} T^{37} + 350408 p^{14} T^{38} + 49416 p^{16} T^{39} + 122763 p^{16} T^{40} + 10104 p^{18} T^{41} + 18074 p^{18} T^{42} + 1132 p^{20} T^{43} + 1543 p^{20} T^{44} + 72 p^{22} T^{45} + 69 p^{22} T^{46} + 2 p^{24} T^{47} + p^{24} T^{48} \)
7 \( 1 + 8 T + 47 T^{2} + 44 p T^{3} + 1588 T^{4} + 7360 T^{5} + 34166 T^{6} + 144604 T^{7} + 577659 T^{8} + 2264690 T^{9} + 8457096 T^{10} + 30344066 T^{11} + 106479519 T^{12} + 362138522 T^{13} + 1193861612 T^{14} + 3851358892 T^{15} + 12137750496 T^{16} + 37275570208 T^{17} + 112141246748 T^{18} + 330885983760 T^{19} + 954502527463 T^{20} + 2699186979130 T^{21} + 7500422996193 T^{22} + 20414790798246 T^{23} + 54471638150829 T^{24} + 20414790798246 p T^{25} + 7500422996193 p^{2} T^{26} + 2699186979130 p^{3} T^{27} + 954502527463 p^{4} T^{28} + 330885983760 p^{5} T^{29} + 112141246748 p^{6} T^{30} + 37275570208 p^{7} T^{31} + 12137750496 p^{8} T^{32} + 3851358892 p^{9} T^{33} + 1193861612 p^{10} T^{34} + 362138522 p^{11} T^{35} + 106479519 p^{12} T^{36} + 30344066 p^{13} T^{37} + 8457096 p^{14} T^{38} + 2264690 p^{15} T^{39} + 577659 p^{16} T^{40} + 144604 p^{17} T^{41} + 34166 p^{18} T^{42} + 7360 p^{19} T^{43} + 1588 p^{20} T^{44} + 44 p^{22} T^{45} + 47 p^{22} T^{46} + 8 p^{23} T^{47} + p^{24} T^{48} \)
11 \( 1 + 16 T + 108 T^{2} + 482 T^{3} + 200 p T^{4} + 9936 T^{5} + 33873 T^{6} + 82034 T^{7} + 81171 T^{8} - 559980 T^{9} - 4184272 T^{10} - 22108264 T^{11} - 99945410 T^{12} - 306135694 T^{13} - 627800152 T^{14} - 730008218 T^{15} + 307336305 p T^{16} + 28703101904 T^{17} + 123571834060 T^{18} + 444902165888 T^{19} + 1085672629627 T^{20} + 984138768406 T^{21} - 2944397444687 T^{22} - 27703297781850 T^{23} - 129366131631747 T^{24} - 27703297781850 p T^{25} - 2944397444687 p^{2} T^{26} + 984138768406 p^{3} T^{27} + 1085672629627 p^{4} T^{28} + 444902165888 p^{5} T^{29} + 123571834060 p^{6} T^{30} + 28703101904 p^{7} T^{31} + 307336305 p^{9} T^{32} - 730008218 p^{9} T^{33} - 627800152 p^{10} T^{34} - 306135694 p^{11} T^{35} - 99945410 p^{12} T^{36} - 22108264 p^{13} T^{37} - 4184272 p^{14} T^{38} - 559980 p^{15} T^{39} + 81171 p^{16} T^{40} + 82034 p^{17} T^{41} + 33873 p^{18} T^{42} + 9936 p^{19} T^{43} + 200 p^{21} T^{44} + 482 p^{21} T^{45} + 108 p^{22} T^{46} + 16 p^{23} T^{47} + p^{24} T^{48} \)
13 \( 1 + 35 T^{2} - 162 T^{3} + 605 T^{4} - 6380 T^{5} + 18417 T^{6} - 122060 T^{7} + 560195 T^{8} - 1987984 T^{9} + 10934860 T^{10} - 36695260 T^{11} + 154920543 T^{12} - 639987630 T^{13} + 2053146690 T^{14} - 8676406446 T^{15} + 2293360340 p T^{16} - 99128662140 T^{17} + 403319750114 T^{18} - 1211549670290 T^{19} + 4684861728995 T^{20} - 16308994332932 T^{21} + 53717124839710 T^{22} - 214008855453230 T^{23} + 672464802882643 T^{24} - 214008855453230 p T^{25} + 53717124839710 p^{2} T^{26} - 16308994332932 p^{3} T^{27} + 4684861728995 p^{4} T^{28} - 1211549670290 p^{5} T^{29} + 403319750114 p^{6} T^{30} - 99128662140 p^{7} T^{31} + 2293360340 p^{9} T^{32} - 8676406446 p^{9} T^{33} + 2053146690 p^{10} T^{34} - 639987630 p^{11} T^{35} + 154920543 p^{12} T^{36} - 36695260 p^{13} T^{37} + 10934860 p^{14} T^{38} - 1987984 p^{15} T^{39} + 560195 p^{16} T^{40} - 122060 p^{17} T^{41} + 18417 p^{18} T^{42} - 6380 p^{19} T^{43} + 605 p^{20} T^{44} - 162 p^{21} T^{45} + 35 p^{22} T^{46} + p^{24} T^{48} \)
17 \( 1 - 8 T + 12 T^{2} + 250 T^{3} - 1464 T^{4} + 2432 T^{5} + 14062 T^{6} - 117720 T^{7} + 614496 T^{8} - 1611598 T^{9} - 5795888 T^{10} + 64294880 T^{11} - 255643609 T^{12} + 370469872 T^{13} + 1146916432 T^{14} - 19163047230 T^{15} + 107061669956 T^{16} - 301259801808 T^{17} - 335731156448 T^{18} + 6962302063320 T^{19} - 36089636110584 T^{20} + 111892533373632 T^{21} + 5681586321912 T^{22} - 2030942416645080 T^{23} + 12882219967505801 T^{24} - 2030942416645080 p T^{25} + 5681586321912 p^{2} T^{26} + 111892533373632 p^{3} T^{27} - 36089636110584 p^{4} T^{28} + 6962302063320 p^{5} T^{29} - 335731156448 p^{6} T^{30} - 301259801808 p^{7} T^{31} + 107061669956 p^{8} T^{32} - 19163047230 p^{9} T^{33} + 1146916432 p^{10} T^{34} + 370469872 p^{11} T^{35} - 255643609 p^{12} T^{36} + 64294880 p^{13} T^{37} - 5795888 p^{14} T^{38} - 1611598 p^{15} T^{39} + 614496 p^{16} T^{40} - 117720 p^{17} T^{41} + 14062 p^{18} T^{42} + 2432 p^{19} T^{43} - 1464 p^{20} T^{44} + 250 p^{21} T^{45} + 12 p^{22} T^{46} - 8 p^{23} T^{47} + p^{24} T^{48} \)
19 \( 1 - 16 T + 198 T^{2} - 1856 T^{3} + 13950 T^{4} - 90776 T^{5} + 506374 T^{6} - 2460764 T^{7} + 10565553 T^{8} - 38542084 T^{9} + 112379322 T^{10} - 218525372 T^{11} - 191463132 T^{12} + 3784143244 T^{13} - 15986642156 T^{14} + 23720769684 T^{15} + 248369122612 T^{16} - 2416997375540 T^{17} + 11338675056474 T^{18} - 27790891303972 T^{19} - 92308623023958 T^{20} + 1700893394808972 T^{21} - 12786353964772772 T^{22} + 73199443385533440 T^{23} - 350106543514634436 T^{24} + 73199443385533440 p T^{25} - 12786353964772772 p^{2} T^{26} + 1700893394808972 p^{3} T^{27} - 92308623023958 p^{4} T^{28} - 27790891303972 p^{5} T^{29} + 11338675056474 p^{6} T^{30} - 2416997375540 p^{7} T^{31} + 248369122612 p^{8} T^{32} + 23720769684 p^{9} T^{33} - 15986642156 p^{10} T^{34} + 3784143244 p^{11} T^{35} - 191463132 p^{12} T^{36} - 218525372 p^{13} T^{37} + 112379322 p^{14} T^{38} - 38542084 p^{15} T^{39} + 10565553 p^{16} T^{40} - 2460764 p^{17} T^{41} + 506374 p^{18} T^{42} - 90776 p^{19} T^{43} + 13950 p^{20} T^{44} - 1856 p^{21} T^{45} + 198 p^{22} T^{46} - 16 p^{23} T^{47} + p^{24} T^{48} \)
23 \( 1 - 12 T - 3 p T^{2} + 1644 T^{3} - 147 p T^{4} - 81980 T^{5} + 558028 T^{6} + 815644 T^{7} - 24556457 T^{8} + 4019924 p T^{9} + 300556492 T^{10} - 4248723460 T^{11} + 13471551219 T^{12} + 1948308924 p T^{13} - 584927415576 T^{14} + 1929517003416 T^{15} + 4418815587486 T^{16} - 69689603378100 T^{17} + 247685586175302 T^{18} + 331489611583716 T^{19} - 6653264718841433 T^{20} + 25343508755353048 T^{21} - 7552484126591187 T^{22} - 397456451526795220 T^{23} + 2584361127218239831 T^{24} - 397456451526795220 p T^{25} - 7552484126591187 p^{2} T^{26} + 25343508755353048 p^{3} T^{27} - 6653264718841433 p^{4} T^{28} + 331489611583716 p^{5} T^{29} + 247685586175302 p^{6} T^{30} - 69689603378100 p^{7} T^{31} + 4418815587486 p^{8} T^{32} + 1929517003416 p^{9} T^{33} - 584927415576 p^{10} T^{34} + 1948308924 p^{12} T^{35} + 13471551219 p^{12} T^{36} - 4248723460 p^{13} T^{37} + 300556492 p^{14} T^{38} + 4019924 p^{16} T^{39} - 24556457 p^{16} T^{40} + 815644 p^{17} T^{41} + 558028 p^{18} T^{42} - 81980 p^{19} T^{43} - 147 p^{21} T^{44} + 1644 p^{21} T^{45} - 3 p^{23} T^{46} - 12 p^{23} T^{47} + p^{24} T^{48} \)
29 \( 1 - 40 T + 885 T^{2} - 13842 T^{3} + 169020 T^{4} - 1697280 T^{5} + 14488017 T^{6} - 107831890 T^{7} + 717839155 T^{8} - 4412789018 T^{9} + 26108898135 T^{10} - 155156048930 T^{11} + 947280512417 T^{12} - 5883075309380 T^{13} + 36173067728540 T^{14} - 215900429213890 T^{15} + 1249093588777930 T^{16} - 7101950701798090 T^{17} + 40376935853282833 T^{18} - 231335264752931780 T^{19} + 1327441309253073165 T^{20} - 7535999561902751674 T^{21} + 42004057092215654420 T^{22} - \)\(23\!\cdots\!40\)\( T^{23} + \)\(12\!\cdots\!05\)\( T^{24} - \)\(23\!\cdots\!40\)\( p T^{25} + 42004057092215654420 p^{2} T^{26} - 7535999561902751674 p^{3} T^{27} + 1327441309253073165 p^{4} T^{28} - 231335264752931780 p^{5} T^{29} + 40376935853282833 p^{6} T^{30} - 7101950701798090 p^{7} T^{31} + 1249093588777930 p^{8} T^{32} - 215900429213890 p^{9} T^{33} + 36173067728540 p^{10} T^{34} - 5883075309380 p^{11} T^{35} + 947280512417 p^{12} T^{36} - 155156048930 p^{13} T^{37} + 26108898135 p^{14} T^{38} - 4412789018 p^{15} T^{39} + 717839155 p^{16} T^{40} - 107831890 p^{17} T^{41} + 14488017 p^{18} T^{42} - 1697280 p^{19} T^{43} + 169020 p^{20} T^{44} - 13842 p^{21} T^{45} + 885 p^{22} T^{46} - 40 p^{23} T^{47} + p^{24} T^{48} \)
31 \( 1 + 12 T - 58 T^{2} - 828 T^{3} + 5393 T^{4} + 47508 T^{5} - 239472 T^{6} - 1258640 T^{7} + 9788058 T^{8} + 36598572 T^{9} - 99812186 T^{10} - 724525152 T^{11} - 3647006766 T^{12} + 50197245840 T^{13} + 388048308242 T^{14} - 1905231243964 T^{15} - 11148079085201 T^{16} + 80741697278812 T^{17} + 372046651216646 T^{18} - 1606033991865552 T^{19} - 7395264303345986 T^{20} + 40615791373800800 T^{21} + 303272802849091438 T^{22} - 51905722083914748 T^{23} - 6545307433662486577 T^{24} - 51905722083914748 p T^{25} + 303272802849091438 p^{2} T^{26} + 40615791373800800 p^{3} T^{27} - 7395264303345986 p^{4} T^{28} - 1606033991865552 p^{5} T^{29} + 372046651216646 p^{6} T^{30} + 80741697278812 p^{7} T^{31} - 11148079085201 p^{8} T^{32} - 1905231243964 p^{9} T^{33} + 388048308242 p^{10} T^{34} + 50197245840 p^{11} T^{35} - 3647006766 p^{12} T^{36} - 724525152 p^{13} T^{37} - 99812186 p^{14} T^{38} + 36598572 p^{15} T^{39} + 9788058 p^{16} T^{40} - 1258640 p^{17} T^{41} - 239472 p^{18} T^{42} + 47508 p^{19} T^{43} + 5393 p^{20} T^{44} - 828 p^{21} T^{45} - 58 p^{22} T^{46} + 12 p^{23} T^{47} + p^{24} T^{48} \)
37 \( 1 - 31 T^{2} - 352 T^{3} + 3810 T^{4} - 8512 T^{5} - 91828 T^{6} - 296872 T^{7} + 12215343 T^{8} - 35554048 T^{9} - 196874836 T^{10} + 1128372792 T^{11} + 15642254118 T^{12} - 137909126408 T^{13} - 6698649245 p T^{14} + 3842168195152 T^{15} + 4377942560548 T^{16} - 238611085698672 T^{17} + 883447852888759 T^{18} + 6379719134991736 T^{19} - 21227559925662933 T^{20} - 274841639156562872 T^{21} + 2387913792251587045 T^{22} - 2071935967324534920 T^{23} - 60670844809628182518 T^{24} - 2071935967324534920 p T^{25} + 2387913792251587045 p^{2} T^{26} - 274841639156562872 p^{3} T^{27} - 21227559925662933 p^{4} T^{28} + 6379719134991736 p^{5} T^{29} + 883447852888759 p^{6} T^{30} - 238611085698672 p^{7} T^{31} + 4377942560548 p^{8} T^{32} + 3842168195152 p^{9} T^{33} - 6698649245 p^{11} T^{34} - 137909126408 p^{11} T^{35} + 15642254118 p^{12} T^{36} + 1128372792 p^{13} T^{37} - 196874836 p^{14} T^{38} - 35554048 p^{15} T^{39} + 12215343 p^{16} T^{40} - 296872 p^{17} T^{41} - 91828 p^{18} T^{42} - 8512 p^{19} T^{43} + 3810 p^{20} T^{44} - 352 p^{21} T^{45} - 31 p^{22} T^{46} + p^{24} T^{48} \)
43 \( 1 + 47 T^{2} + 240 T^{3} + 5511 T^{4} + 11280 T^{5} + 360736 T^{6} + 23160 p T^{7} + 21457647 T^{8} + 78343540 T^{9} + 1291842392 T^{10} + 2502392040 T^{11} + 70224497891 T^{12} + 203819945520 T^{13} + 3341209992188 T^{14} + 6643696211460 T^{15} + 170756618303902 T^{16} + 317714712905620 T^{17} + 7630569116649706 T^{18} + 14172723048042120 T^{19} + 337169247736344123 T^{20} + 453783479882066420 T^{21} + 16148832035324618001 T^{22} + 21712663220957446340 T^{23} + \)\(62\!\cdots\!91\)\( T^{24} + 21712663220957446340 p T^{25} + 16148832035324618001 p^{2} T^{26} + 453783479882066420 p^{3} T^{27} + 337169247736344123 p^{4} T^{28} + 14172723048042120 p^{5} T^{29} + 7630569116649706 p^{6} T^{30} + 317714712905620 p^{7} T^{31} + 170756618303902 p^{8} T^{32} + 6643696211460 p^{9} T^{33} + 3341209992188 p^{10} T^{34} + 203819945520 p^{11} T^{35} + 70224497891 p^{12} T^{36} + 2502392040 p^{13} T^{37} + 1291842392 p^{14} T^{38} + 78343540 p^{15} T^{39} + 21457647 p^{16} T^{40} + 23160 p^{18} T^{41} + 360736 p^{18} T^{42} + 11280 p^{19} T^{43} + 5511 p^{20} T^{44} + 240 p^{21} T^{45} + 47 p^{22} T^{46} + p^{24} T^{48} \)
47 \( 1 + 12 T + 322 T^{2} + 2872 T^{3} + 50912 T^{4} + 388180 T^{5} + 5523568 T^{6} + 37724460 T^{7} + 462633260 T^{8} + 2933841004 T^{9} + 31522378032 T^{10} + 186923803296 T^{11} + 1758882072011 T^{12} + 9637553744536 T^{13} + 78645212307888 T^{14} + 372175814876036 T^{15} + 2547165141036316 T^{16} + 7367144146142992 T^{17} + 31671952929071988 T^{18} - 334509837050584904 T^{19} - 3081500027795248300 T^{20} - 47416006421771659220 T^{21} - \)\(31\!\cdots\!42\)\( T^{22} - \)\(31\!\cdots\!72\)\( T^{23} - \)\(17\!\cdots\!39\)\( T^{24} - \)\(31\!\cdots\!72\)\( p T^{25} - \)\(31\!\cdots\!42\)\( p^{2} T^{26} - 47416006421771659220 p^{3} T^{27} - 3081500027795248300 p^{4} T^{28} - 334509837050584904 p^{5} T^{29} + 31671952929071988 p^{6} T^{30} + 7367144146142992 p^{7} T^{31} + 2547165141036316 p^{8} T^{32} + 372175814876036 p^{9} T^{33} + 78645212307888 p^{10} T^{34} + 9637553744536 p^{11} T^{35} + 1758882072011 p^{12} T^{36} + 186923803296 p^{13} T^{37} + 31522378032 p^{14} T^{38} + 2933841004 p^{15} T^{39} + 462633260 p^{16} T^{40} + 37724460 p^{17} T^{41} + 5523568 p^{18} T^{42} + 388180 p^{19} T^{43} + 50912 p^{20} T^{44} + 2872 p^{21} T^{45} + 322 p^{22} T^{46} + 12 p^{23} T^{47} + p^{24} T^{48} \)
53 \( 1 + 26 T + 308 T^{2} + 950 T^{3} - 26734 T^{4} - 479364 T^{5} - 3595532 T^{6} - 1892270 T^{7} + 276338341 T^{8} + 3349411596 T^{9} + 17840150788 T^{10} - 35606629520 T^{11} - 1588370775664 T^{12} - 14634239766314 T^{13} - 53284287185632 T^{14} + 354308618346370 T^{15} + 6919019876199376 T^{16} + 49550081590116016 T^{17} + 97123727618756668 T^{18} - 1899199833999411900 T^{19} - 24969747017298863814 T^{20} - \)\(14\!\cdots\!24\)\( T^{21} - 84560812000975570792 T^{22} + \)\(71\!\cdots\!70\)\( T^{23} + \)\(76\!\cdots\!36\)\( T^{24} + \)\(71\!\cdots\!70\)\( p T^{25} - 84560812000975570792 p^{2} T^{26} - \)\(14\!\cdots\!24\)\( p^{3} T^{27} - 24969747017298863814 p^{4} T^{28} - 1899199833999411900 p^{5} T^{29} + 97123727618756668 p^{6} T^{30} + 49550081590116016 p^{7} T^{31} + 6919019876199376 p^{8} T^{32} + 354308618346370 p^{9} T^{33} - 53284287185632 p^{10} T^{34} - 14634239766314 p^{11} T^{35} - 1588370775664 p^{12} T^{36} - 35606629520 p^{13} T^{37} + 17840150788 p^{14} T^{38} + 3349411596 p^{15} T^{39} + 276338341 p^{16} T^{40} - 1892270 p^{17} T^{41} - 3595532 p^{18} T^{42} - 479364 p^{19} T^{43} - 26734 p^{20} T^{44} + 950 p^{21} T^{45} + 308 p^{22} T^{46} + 26 p^{23} T^{47} + p^{24} T^{48} \)
59 \( 1 - 6 T - 79 T^{2} - 854 T^{3} + 9149 T^{4} + 62214 T^{5} + 817000 T^{6} - 6759212 T^{7} - 51976479 T^{8} - 644955324 T^{9} + 5763614408 T^{10} + 27982948186 T^{11} + 323797131125 T^{12} - 3743688112592 T^{13} - 8420268642180 T^{14} - 157538490943580 T^{15} + 2078495662014088 T^{16} + 1423367057738890 T^{17} + 69953469142583626 T^{18} - 1053257891707394100 T^{19} + 1671774578760154721 T^{20} - 29334141078443702350 T^{21} + \)\(39\!\cdots\!95\)\( T^{22} - \)\(20\!\cdots\!88\)\( T^{23} + \)\(14\!\cdots\!91\)\( T^{24} - \)\(20\!\cdots\!88\)\( p T^{25} + \)\(39\!\cdots\!95\)\( p^{2} T^{26} - 29334141078443702350 p^{3} T^{27} + 1671774578760154721 p^{4} T^{28} - 1053257891707394100 p^{5} T^{29} + 69953469142583626 p^{6} T^{30} + 1423367057738890 p^{7} T^{31} + 2078495662014088 p^{8} T^{32} - 157538490943580 p^{9} T^{33} - 8420268642180 p^{10} T^{34} - 3743688112592 p^{11} T^{35} + 323797131125 p^{12} T^{36} + 27982948186 p^{13} T^{37} + 5763614408 p^{14} T^{38} - 644955324 p^{15} T^{39} - 51976479 p^{16} T^{40} - 6759212 p^{17} T^{41} + 817000 p^{18} T^{42} + 62214 p^{19} T^{43} + 9149 p^{20} T^{44} - 854 p^{21} T^{45} - 79 p^{22} T^{46} - 6 p^{23} T^{47} + p^{24} T^{48} \)
61 \( 1 - 30 T + 643 T^{2} - 8780 T^{3} + 103945 T^{4} - 959420 T^{5} + 9594326 T^{6} - 88107910 T^{7} + 965913213 T^{8} - 8605462900 T^{9} + 81164296068 T^{10} - 564175237660 T^{11} + 4658190442633 T^{12} - 28066738182950 T^{13} + 4240386142914 p T^{14} - 1351676263672100 T^{15} + 11389904012090720 T^{16} - 8152674653271830 T^{17} + 16394543225451454 T^{18} + 4987273130964409950 T^{19} - 28323289250241196743 T^{20} + \)\(49\!\cdots\!30\)\( T^{21} - \)\(30\!\cdots\!25\)\( T^{22} + \)\(42\!\cdots\!50\)\( T^{23} - \)\(24\!\cdots\!77\)\( T^{24} + \)\(42\!\cdots\!50\)\( p T^{25} - \)\(30\!\cdots\!25\)\( p^{2} T^{26} + \)\(49\!\cdots\!30\)\( p^{3} T^{27} - 28323289250241196743 p^{4} T^{28} + 4987273130964409950 p^{5} T^{29} + 16394543225451454 p^{6} T^{30} - 8152674653271830 p^{7} T^{31} + 11389904012090720 p^{8} T^{32} - 1351676263672100 p^{9} T^{33} + 4240386142914 p^{11} T^{34} - 28066738182950 p^{11} T^{35} + 4658190442633 p^{12} T^{36} - 564175237660 p^{13} T^{37} + 81164296068 p^{14} T^{38} - 8605462900 p^{15} T^{39} + 965913213 p^{16} T^{40} - 88107910 p^{17} T^{41} + 9594326 p^{18} T^{42} - 959420 p^{19} T^{43} + 103945 p^{20} T^{44} - 8780 p^{21} T^{45} + 643 p^{22} T^{46} - 30 p^{23} T^{47} + p^{24} T^{48} \)
67 \( 1 + 22 T + 192 T^{2} + 1212 T^{3} + 6532 T^{4} - 20060 T^{5} - 612752 T^{6} - 7603380 T^{7} - 91364365 T^{8} - 590833536 T^{9} - 2247178708 T^{10} - 2745454534 T^{11} + 216217496116 T^{12} + 2710661468376 T^{13} + 18266688926088 T^{14} + 1673530476048 p T^{15} + 347623217136516 T^{16} + 715638468896542 T^{17} + 2841115146010548 T^{18} - 208833468612881424 T^{19} - 2382870457635750180 T^{20} - 16616012878687014030 T^{21} - \)\(15\!\cdots\!92\)\( T^{22} - \)\(13\!\cdots\!72\)\( T^{23} - \)\(10\!\cdots\!44\)\( T^{24} - \)\(13\!\cdots\!72\)\( p T^{25} - \)\(15\!\cdots\!92\)\( p^{2} T^{26} - 16616012878687014030 p^{3} T^{27} - 2382870457635750180 p^{4} T^{28} - 208833468612881424 p^{5} T^{29} + 2841115146010548 p^{6} T^{30} + 715638468896542 p^{7} T^{31} + 347623217136516 p^{8} T^{32} + 1673530476048 p^{10} T^{33} + 18266688926088 p^{10} T^{34} + 2710661468376 p^{11} T^{35} + 216217496116 p^{12} T^{36} - 2745454534 p^{13} T^{37} - 2247178708 p^{14} T^{38} - 590833536 p^{15} T^{39} - 91364365 p^{16} T^{40} - 7603380 p^{17} T^{41} - 612752 p^{18} T^{42} - 20060 p^{19} T^{43} + 6532 p^{20} T^{44} + 1212 p^{21} T^{45} + 192 p^{22} T^{46} + 22 p^{23} T^{47} + p^{24} T^{48} \)
71 \( 1 - 4 T + 258 T^{2} + 268 T^{3} + 20144 T^{4} + 234576 T^{5} + 714656 T^{6} + 21288936 T^{7} + 155959908 T^{8} - 155126280 T^{9} + 18354192328 T^{10} - 38838127352 T^{11} - 69373434569 T^{12} + 4603691552608 T^{13} - 82787428018588 T^{14} - 158046183946432 T^{15} + 557146788520672 T^{16} - 64790237382204132 T^{17} + 214134834379836044 T^{18} - 51991252242689780 T^{19} - 10270988790452718124 T^{20} + \)\(30\!\cdots\!84\)\( T^{21} + \)\(11\!\cdots\!22\)\( T^{22} + \)\(56\!\cdots\!88\)\( T^{23} + \)\(26\!\cdots\!77\)\( T^{24} + \)\(56\!\cdots\!88\)\( p T^{25} + \)\(11\!\cdots\!22\)\( p^{2} T^{26} + \)\(30\!\cdots\!84\)\( p^{3} T^{27} - 10270988790452718124 p^{4} T^{28} - 51991252242689780 p^{5} T^{29} + 214134834379836044 p^{6} T^{30} - 64790237382204132 p^{7} T^{31} + 557146788520672 p^{8} T^{32} - 158046183946432 p^{9} T^{33} - 82787428018588 p^{10} T^{34} + 4603691552608 p^{11} T^{35} - 69373434569 p^{12} T^{36} - 38838127352 p^{13} T^{37} + 18354192328 p^{14} T^{38} - 155126280 p^{15} T^{39} + 155959908 p^{16} T^{40} + 21288936 p^{17} T^{41} + 714656 p^{18} T^{42} + 234576 p^{19} T^{43} + 20144 p^{20} T^{44} + 268 p^{21} T^{45} + 258 p^{22} T^{46} - 4 p^{23} T^{47} + p^{24} T^{48} \)
73 \( 1 - 874 T^{2} + 391171 T^{4} - 118947082 T^{6} + 27516572595 T^{8} - 5145325978708 T^{10} + 807412482843850 T^{12} - 109043407326620798 T^{14} + 12902445992963232355 T^{16} - \)\(13\!\cdots\!22\)\( T^{18} + \)\(12\!\cdots\!99\)\( T^{20} - \)\(10\!\cdots\!44\)\( T^{22} + \)\(82\!\cdots\!59\)\( T^{24} - \)\(10\!\cdots\!44\)\( p^{2} T^{26} + \)\(12\!\cdots\!99\)\( p^{4} T^{28} - \)\(13\!\cdots\!22\)\( p^{6} T^{30} + 12902445992963232355 p^{8} T^{32} - 109043407326620798 p^{10} T^{34} + 807412482843850 p^{12} T^{36} - 5145325978708 p^{14} T^{38} + 27516572595 p^{16} T^{40} - 118947082 p^{18} T^{42} + 391171 p^{20} T^{44} - 874 p^{22} T^{46} + p^{24} T^{48} \)
79 \( 1 + 2 T + 2 T^{2} - 868 T^{3} - 22311 T^{4} - 45746 T^{5} + 329842 T^{6} + 14522686 T^{7} + 290610050 T^{8} + 400199680 T^{9} - 3981059068 T^{10} - 161106953084 T^{11} - 2670559301755 T^{12} - 3240647372500 T^{13} + 41170519923192 T^{14} + 1263396020978966 T^{15} + 20178683052160472 T^{16} + 25465462088148412 T^{17} - 241079467534751210 T^{18} - 7668141687095625144 T^{19} - \)\(13\!\cdots\!78\)\( T^{20} - \)\(29\!\cdots\!02\)\( T^{21} + \)\(57\!\cdots\!22\)\( T^{22} + \)\(37\!\cdots\!10\)\( T^{23} + \)\(87\!\cdots\!98\)\( T^{24} + \)\(37\!\cdots\!10\)\( p T^{25} + \)\(57\!\cdots\!22\)\( p^{2} T^{26} - \)\(29\!\cdots\!02\)\( p^{3} T^{27} - \)\(13\!\cdots\!78\)\( p^{4} T^{28} - 7668141687095625144 p^{5} T^{29} - 241079467534751210 p^{6} T^{30} + 25465462088148412 p^{7} T^{31} + 20178683052160472 p^{8} T^{32} + 1263396020978966 p^{9} T^{33} + 41170519923192 p^{10} T^{34} - 3240647372500 p^{11} T^{35} - 2670559301755 p^{12} T^{36} - 161106953084 p^{13} T^{37} - 3981059068 p^{14} T^{38} + 400199680 p^{15} T^{39} + 290610050 p^{16} T^{40} + 14522686 p^{17} T^{41} + 329842 p^{18} T^{42} - 45746 p^{19} T^{43} - 22311 p^{20} T^{44} - 868 p^{21} T^{45} + 2 p^{22} T^{46} + 2 p^{23} T^{47} + p^{24} T^{48} \)
83 \( ( 1 - 40 T + 1370 T^{2} - 31604 T^{3} + 649745 T^{4} - 10912580 T^{5} + 168094004 T^{6} - 2262435440 T^{7} + 28486604420 T^{8} - 324005854908 T^{9} + 3495017102130 T^{10} - 34658870817460 T^{11} + 328380939306004 T^{12} - 34658870817460 p T^{13} + 3495017102130 p^{2} T^{14} - 324005854908 p^{3} T^{15} + 28486604420 p^{4} T^{16} - 2262435440 p^{5} T^{17} + 168094004 p^{6} T^{18} - 10912580 p^{7} T^{19} + 649745 p^{8} T^{20} - 31604 p^{9} T^{21} + 1370 p^{10} T^{22} - 40 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
89 \( 1 + 72 T + 2567 T^{2} + 60970 T^{3} + 1092241 T^{4} + 15660888 T^{5} + 182069669 T^{6} + 1640794008 T^{7} + 9271904375 T^{8} - 19052101824 T^{9} - 1369188376120 T^{10} - 22109654039672 T^{11} - 245063159722697 T^{12} - 2075112846821886 T^{13} - 13379750973521818 T^{14} - 63746136727123158 T^{15} - 289078453261688524 T^{16} - 3434221997490479008 T^{17} - 58230569487852996538 T^{18} - \)\(67\!\cdots\!62\)\( T^{19} - \)\(37\!\cdots\!73\)\( T^{20} + \)\(32\!\cdots\!56\)\( T^{21} + \)\(11\!\cdots\!70\)\( T^{22} + \)\(18\!\cdots\!42\)\( T^{23} + \)\(19\!\cdots\!95\)\( T^{24} + \)\(18\!\cdots\!42\)\( p T^{25} + \)\(11\!\cdots\!70\)\( p^{2} T^{26} + \)\(32\!\cdots\!56\)\( p^{3} T^{27} - \)\(37\!\cdots\!73\)\( p^{4} T^{28} - \)\(67\!\cdots\!62\)\( p^{5} T^{29} - 58230569487852996538 p^{6} T^{30} - 3434221997490479008 p^{7} T^{31} - 289078453261688524 p^{8} T^{32} - 63746136727123158 p^{9} T^{33} - 13379750973521818 p^{10} T^{34} - 2075112846821886 p^{11} T^{35} - 245063159722697 p^{12} T^{36} - 22109654039672 p^{13} T^{37} - 1369188376120 p^{14} T^{38} - 19052101824 p^{15} T^{39} + 9271904375 p^{16} T^{40} + 1640794008 p^{17} T^{41} + 182069669 p^{18} T^{42} + 15660888 p^{19} T^{43} + 1092241 p^{20} T^{44} + 60970 p^{21} T^{45} + 2567 p^{22} T^{46} + 72 p^{23} T^{47} + p^{24} T^{48} \)
97 \( 1 + 22 T + 107 T^{2} - 3040 T^{3} - 55931 T^{4} - 272808 T^{5} + 3812971 T^{6} + 81333966 T^{7} + 595726617 T^{8} - 3106418368 T^{9} - 118438577172 T^{10} - 1082504874432 T^{11} + 1218256489548 T^{12} + 131529893358024 T^{13} + 1378527839988420 T^{14} + 3035821796559608 T^{15} - 97822009229839764 T^{16} - 1511308395214441552 T^{17} - 8912531893145600212 T^{18} + 44386565707342981216 T^{19} + \)\(14\!\cdots\!26\)\( T^{20} + \)\(12\!\cdots\!92\)\( T^{21} + \)\(17\!\cdots\!14\)\( T^{22} - \)\(85\!\cdots\!28\)\( T^{23} - \)\(12\!\cdots\!58\)\( T^{24} - \)\(85\!\cdots\!28\)\( p T^{25} + \)\(17\!\cdots\!14\)\( p^{2} T^{26} + \)\(12\!\cdots\!92\)\( p^{3} T^{27} + \)\(14\!\cdots\!26\)\( p^{4} T^{28} + 44386565707342981216 p^{5} T^{29} - 8912531893145600212 p^{6} T^{30} - 1511308395214441552 p^{7} T^{31} - 97822009229839764 p^{8} T^{32} + 3035821796559608 p^{9} T^{33} + 1378527839988420 p^{10} T^{34} + 131529893358024 p^{11} T^{35} + 1218256489548 p^{12} T^{36} - 1082504874432 p^{13} T^{37} - 118438577172 p^{14} T^{38} - 3106418368 p^{15} T^{39} + 595726617 p^{16} T^{40} + 81333966 p^{17} T^{41} + 3812971 p^{18} T^{42} - 272808 p^{19} T^{43} - 55931 p^{20} T^{44} - 3040 p^{21} T^{45} + 107 p^{22} T^{46} + 22 p^{23} T^{47} + p^{24} T^{48} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.60918941050621403651065942239, −4.58568720787821748340612848802, −4.52866666992866491083707694709, −4.41062707061487130824974387030, −4.39929894846140477493809162559, −3.99963340873282751388752589139, −3.98509778115369306253567560798, −3.79708842512108554983900580860, −3.64536165846230725817615291892, −3.63319350333662729485157751021, −3.60681902096579331110948601170, −3.41914894908407211552846829199, −3.30540078809108426168597042181, −3.22065244054584463765262350270, −3.14904209337022793535857170845, −3.13521749474630646733837386435, −3.11011041795528540377520344850, −3.10376594578513509992757684009, −2.65731920454566601151662690035, −2.60491280530989260121374932466, −2.44489912598634372225288123875, −2.12528537002233437980815050805, −1.91115962881095603746001628930, −1.06504499463299459179779644215, −0.67873017047084977318755823106, 0.67873017047084977318755823106, 1.06504499463299459179779644215, 1.91115962881095603746001628930, 2.12528537002233437980815050805, 2.44489912598634372225288123875, 2.60491280530989260121374932466, 2.65731920454566601151662690035, 3.10376594578513509992757684009, 3.11011041795528540377520344850, 3.13521749474630646733837386435, 3.14904209337022793535857170845, 3.22065244054584463765262350270, 3.30540078809108426168597042181, 3.41914894908407211552846829199, 3.60681902096579331110948601170, 3.63319350333662729485157751021, 3.64536165846230725817615291892, 3.79708842512108554983900580860, 3.98509778115369306253567560798, 3.99963340873282751388752589139, 4.39929894846140477493809162559, 4.41062707061487130824974387030, 4.52866666992866491083707694709, 4.58568720787821748340612848802, 4.60918941050621403651065942239

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.