Properties

Degree 24
Conductor $ 2^{12} \cdot 3^{36} \cdot 149^{12} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·2-s + 78·4-s + 5·5-s − 6·7-s − 364·8-s − 60·10-s + 10·11-s − 13-s + 72·14-s + 1.36e3·16-s + 6·17-s − 10·19-s + 390·20-s − 120·22-s + 15·23-s − 14·25-s + 12·26-s − 468·28-s + 33·29-s − 6·31-s − 4.36e3·32-s − 72·34-s − 30·35-s − 13·37-s + 120·38-s − 1.82e3·40-s + 20·41-s + ⋯
L(s)  = 1  − 8.48·2-s + 39·4-s + 2.23·5-s − 2.26·7-s − 128.·8-s − 18.9·10-s + 3.01·11-s − 0.277·13-s + 19.2·14-s + 341.·16-s + 1.45·17-s − 2.29·19-s + 87.2·20-s − 25.5·22-s + 3.12·23-s − 2.79·25-s + 2.35·26-s − 88.4·28-s + 6.12·29-s − 1.07·31-s − 772.·32-s − 12.3·34-s − 5.07·35-s − 2.13·37-s + 19.4·38-s − 287.·40-s + 3.12·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 149^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 149^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{12} \cdot 3^{36} \cdot 149^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8046} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(24,\ 2^{12} \cdot 3^{36} \cdot 149^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$
$L(1)$  $\approx$  $2.312848896$
$L(\frac12)$  $\approx$  $2.312848896$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;149\}$, \(F_p\) is a polynomial of degree 24. If $p \in \{2,\;3,\;149\}$, then $F_p$ is a polynomial of degree at most 23.
$p$$F_p$
bad2 \( ( 1 + T )^{12} \)
3 \( 1 \)
149 \( ( 1 - T )^{12} \)
good5 \( 1 - p T + 39 T^{2} - 159 T^{3} + 706 T^{4} - 2429 T^{5} + 8052 T^{6} - 23802 T^{7} + 66268 T^{8} - 171524 T^{9} + 429546 T^{10} - 39983 p^{2} T^{11} + 464937 p T^{12} - 39983 p^{3} T^{13} + 429546 p^{2} T^{14} - 171524 p^{3} T^{15} + 66268 p^{4} T^{16} - 23802 p^{5} T^{17} + 8052 p^{6} T^{18} - 2429 p^{7} T^{19} + 706 p^{8} T^{20} - 159 p^{9} T^{21} + 39 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \)
7 \( 1 + 6 T + 59 T^{2} + 248 T^{3} + 1398 T^{4} + 4348 T^{5} + 365 p^{2} T^{6} + 40132 T^{7} + 19073 p T^{8} + 179525 T^{9} + 588556 T^{10} + 128130 T^{11} + 2459561 T^{12} + 128130 p T^{13} + 588556 p^{2} T^{14} + 179525 p^{3} T^{15} + 19073 p^{5} T^{16} + 40132 p^{5} T^{17} + 365 p^{8} T^{18} + 4348 p^{7} T^{19} + 1398 p^{8} T^{20} + 248 p^{9} T^{21} + 59 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 - 10 T + 115 T^{2} - 724 T^{3} + 5023 T^{4} - 24667 T^{5} + 134846 T^{6} - 571308 T^{7} + 2675581 T^{8} - 10038817 T^{9} + 41178796 T^{10} - 12524934 p T^{11} + 503919571 T^{12} - 12524934 p^{2} T^{13} + 41178796 p^{2} T^{14} - 10038817 p^{3} T^{15} + 2675581 p^{4} T^{16} - 571308 p^{5} T^{17} + 134846 p^{6} T^{18} - 24667 p^{7} T^{19} + 5023 p^{8} T^{20} - 724 p^{9} T^{21} + 115 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 + T + 81 T^{2} - 7 T^{3} + 3249 T^{4} - 3204 T^{5} + 89567 T^{6} - 140528 T^{7} + 1937232 T^{8} - 3458175 T^{9} + 34186509 T^{10} - 59592236 T^{11} + 492567871 T^{12} - 59592236 p T^{13} + 34186509 p^{2} T^{14} - 3458175 p^{3} T^{15} + 1937232 p^{4} T^{16} - 140528 p^{5} T^{17} + 89567 p^{6} T^{18} - 3204 p^{7} T^{19} + 3249 p^{8} T^{20} - 7 p^{9} T^{21} + 81 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 - 6 T + 79 T^{2} - 387 T^{3} + 3639 T^{4} - 15822 T^{5} + 119551 T^{6} - 27733 p T^{7} + 3119918 T^{8} - 11308494 T^{9} + 67422746 T^{10} - 226504061 T^{11} + 1241062419 T^{12} - 226504061 p T^{13} + 67422746 p^{2} T^{14} - 11308494 p^{3} T^{15} + 3119918 p^{4} T^{16} - 27733 p^{6} T^{17} + 119551 p^{6} T^{18} - 15822 p^{7} T^{19} + 3639 p^{8} T^{20} - 387 p^{9} T^{21} + 79 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 + 10 T + 127 T^{2} + 934 T^{3} + 6719 T^{4} + 38183 T^{5} + 196112 T^{6} + 876536 T^{7} + 3480005 T^{8} + 12122933 T^{9} + 40733640 T^{10} + 118097532 T^{11} + 509348397 T^{12} + 118097532 p T^{13} + 40733640 p^{2} T^{14} + 12122933 p^{3} T^{15} + 3480005 p^{4} T^{16} + 876536 p^{5} T^{17} + 196112 p^{6} T^{18} + 38183 p^{7} T^{19} + 6719 p^{8} T^{20} + 934 p^{9} T^{21} + 127 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 - 15 T + 268 T^{2} - 2999 T^{3} + 32777 T^{4} - 289696 T^{5} + 2422679 T^{6} - 17672115 T^{7} + 121520833 T^{8} - 751073331 T^{9} + 4381416603 T^{10} - 23264664391 T^{11} + 116738365747 T^{12} - 23264664391 p T^{13} + 4381416603 p^{2} T^{14} - 751073331 p^{3} T^{15} + 121520833 p^{4} T^{16} - 17672115 p^{5} T^{17} + 2422679 p^{6} T^{18} - 289696 p^{7} T^{19} + 32777 p^{8} T^{20} - 2999 p^{9} T^{21} + 268 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 - 33 T + 647 T^{2} - 9211 T^{3} + 106462 T^{4} - 1046559 T^{5} + 9064944 T^{6} - 70513256 T^{7} + 501529378 T^{8} - 3300509040 T^{9} + 20342854914 T^{10} - 118306330217 T^{11} + 653439497197 T^{12} - 118306330217 p T^{13} + 20342854914 p^{2} T^{14} - 3300509040 p^{3} T^{15} + 501529378 p^{4} T^{16} - 70513256 p^{5} T^{17} + 9064944 p^{6} T^{18} - 1046559 p^{7} T^{19} + 106462 p^{8} T^{20} - 9211 p^{9} T^{21} + 647 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 + 6 T + 178 T^{2} + 999 T^{3} + 16153 T^{4} + 86715 T^{5} + 1009493 T^{6} + 5272578 T^{7} + 49204727 T^{8} + 249105881 T^{9} + 1979200121 T^{10} + 9495174993 T^{11} + 66874337227 T^{12} + 9495174993 p T^{13} + 1979200121 p^{2} T^{14} + 249105881 p^{3} T^{15} + 49204727 p^{4} T^{16} + 5272578 p^{5} T^{17} + 1009493 p^{6} T^{18} + 86715 p^{7} T^{19} + 16153 p^{8} T^{20} + 999 p^{9} T^{21} + 178 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 + 13 T + 267 T^{2} + 2513 T^{3} + 31583 T^{4} + 238965 T^{5} + 2355489 T^{6} + 15309362 T^{7} + 131379734 T^{8} + 768853859 T^{9} + 6046087991 T^{10} + 32723836129 T^{11} + 239861350569 T^{12} + 32723836129 p T^{13} + 6046087991 p^{2} T^{14} + 768853859 p^{3} T^{15} + 131379734 p^{4} T^{16} + 15309362 p^{5} T^{17} + 2355489 p^{6} T^{18} + 238965 p^{7} T^{19} + 31583 p^{8} T^{20} + 2513 p^{9} T^{21} + 267 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 - 20 T + 468 T^{2} - 6832 T^{3} + 98146 T^{4} - 1131373 T^{5} + 12475664 T^{6} - 119187864 T^{7} + 1085210964 T^{8} - 8834724827 T^{9} + 68563015800 T^{10} - 482456685988 T^{11} + 3237627435202 T^{12} - 482456685988 p T^{13} + 68563015800 p^{2} T^{14} - 8834724827 p^{3} T^{15} + 1085210964 p^{4} T^{16} - 119187864 p^{5} T^{17} + 12475664 p^{6} T^{18} - 1131373 p^{7} T^{19} + 98146 p^{8} T^{20} - 6832 p^{9} T^{21} + 468 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 + 11 T + 328 T^{2} + 2903 T^{3} + 49571 T^{4} + 361934 T^{5} + 108640 p T^{6} + 29010177 T^{7} + 318352639 T^{8} + 1738315065 T^{9} + 402705963 p T^{10} + 85944319606 T^{11} + 799628157817 T^{12} + 85944319606 p T^{13} + 402705963 p^{3} T^{14} + 1738315065 p^{3} T^{15} + 318352639 p^{4} T^{16} + 29010177 p^{5} T^{17} + 108640 p^{7} T^{18} + 361934 p^{7} T^{19} + 49571 p^{8} T^{20} + 2903 p^{9} T^{21} + 328 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 - 15 T + 516 T^{2} - 6208 T^{3} + 121430 T^{4} - 1225645 T^{5} + 17536021 T^{6} - 151949323 T^{7} + 1743788176 T^{8} - 13120541600 T^{9} + 126243197873 T^{10} - 827657632458 T^{11} + 6834095624191 T^{12} - 827657632458 p T^{13} + 126243197873 p^{2} T^{14} - 13120541600 p^{3} T^{15} + 1743788176 p^{4} T^{16} - 151949323 p^{5} T^{17} + 17536021 p^{6} T^{18} - 1225645 p^{7} T^{19} + 121430 p^{8} T^{20} - 6208 p^{9} T^{21} + 516 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 - 4 T + 317 T^{2} - 1535 T^{3} + 53635 T^{4} - 288929 T^{5} + 6330481 T^{6} - 35530248 T^{7} + 570336844 T^{8} - 3173471249 T^{9} + 41004300501 T^{10} - 216160124956 T^{11} + 2401609762549 T^{12} - 216160124956 p T^{13} + 41004300501 p^{2} T^{14} - 3173471249 p^{3} T^{15} + 570336844 p^{4} T^{16} - 35530248 p^{5} T^{17} + 6330481 p^{6} T^{18} - 288929 p^{7} T^{19} + 53635 p^{8} T^{20} - 1535 p^{9} T^{21} + 317 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 - 10 T + 219 T^{2} - 1772 T^{3} + 32138 T^{4} - 256033 T^{5} + 3645418 T^{6} - 26020690 T^{7} + 326852615 T^{8} - 2230252101 T^{9} + 25077488626 T^{10} - 155431115459 T^{11} + 1578741333065 T^{12} - 155431115459 p T^{13} + 25077488626 p^{2} T^{14} - 2230252101 p^{3} T^{15} + 326852615 p^{4} T^{16} - 26020690 p^{5} T^{17} + 3645418 p^{6} T^{18} - 256033 p^{7} T^{19} + 32138 p^{8} T^{20} - 1772 p^{9} T^{21} + 219 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 + 12 T + 298 T^{2} + 3496 T^{3} + 54064 T^{4} + 546885 T^{5} + 6800109 T^{6} + 62808914 T^{7} + 667720626 T^{8} + 5627341982 T^{9} + 53783116351 T^{10} + 415772734409 T^{11} + 3577595975703 T^{12} + 415772734409 p T^{13} + 53783116351 p^{2} T^{14} + 5627341982 p^{3} T^{15} + 667720626 p^{4} T^{16} + 62808914 p^{5} T^{17} + 6800109 p^{6} T^{18} + 546885 p^{7} T^{19} + 54064 p^{8} T^{20} + 3496 p^{9} T^{21} + 298 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 19 T + 607 T^{2} + 9305 T^{3} + 169538 T^{4} + 2182477 T^{5} + 29437067 T^{6} + 327561454 T^{7} + 3613890498 T^{8} + 35542644025 T^{9} + 337982347007 T^{10} + 2981324477278 T^{11} + 25176664348651 T^{12} + 2981324477278 p T^{13} + 337982347007 p^{2} T^{14} + 35542644025 p^{3} T^{15} + 3613890498 p^{4} T^{16} + 327561454 p^{5} T^{17} + 29437067 p^{6} T^{18} + 2182477 p^{7} T^{19} + 169538 p^{8} T^{20} + 9305 p^{9} T^{21} + 607 p^{10} T^{22} + 19 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 - 47 T + 1415 T^{2} - 30825 T^{3} + 552183 T^{4} - 8406194 T^{5} + 113856814 T^{6} - 1392488028 T^{7} + 15731393240 T^{8} - 164995119135 T^{9} + 1624120581131 T^{10} - 14986823940348 T^{11} + 130317347009003 T^{12} - 14986823940348 p T^{13} + 1624120581131 p^{2} T^{14} - 164995119135 p^{3} T^{15} + 15731393240 p^{4} T^{16} - 1392488028 p^{5} T^{17} + 113856814 p^{6} T^{18} - 8406194 p^{7} T^{19} + 552183 p^{8} T^{20} - 30825 p^{9} T^{21} + 1415 p^{10} T^{22} - 47 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 + 2 T + 486 T^{2} + 1029 T^{3} + 117469 T^{4} + 238412 T^{5} + 18645928 T^{6} + 34124103 T^{7} + 2191985511 T^{8} + 3513470157 T^{9} + 205786863273 T^{10} + 292339731915 T^{11} + 16232626831117 T^{12} + 292339731915 p T^{13} + 205786863273 p^{2} T^{14} + 3513470157 p^{3} T^{15} + 2191985511 p^{4} T^{16} + 34124103 p^{5} T^{17} + 18645928 p^{6} T^{18} + 238412 p^{7} T^{19} + 117469 p^{8} T^{20} + 1029 p^{9} T^{21} + 486 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 + 15 T + 564 T^{2} + 6783 T^{3} + 152314 T^{4} + 1542062 T^{5} + 26410551 T^{6} + 232068714 T^{7} + 3357776408 T^{8} + 26270103298 T^{9} + 339911306201 T^{10} + 2426868197790 T^{11} + 28970653055967 T^{12} + 2426868197790 p T^{13} + 339911306201 p^{2} T^{14} + 26270103298 p^{3} T^{15} + 3357776408 p^{4} T^{16} + 232068714 p^{5} T^{17} + 26410551 p^{6} T^{18} + 1542062 p^{7} T^{19} + 152314 p^{8} T^{20} + 6783 p^{9} T^{21} + 564 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 - 18 T + 863 T^{2} - 13116 T^{3} + 348984 T^{4} - 4560335 T^{5} + 87697456 T^{6} - 996405958 T^{7} + 15277660667 T^{8} - 151747795747 T^{9} + 1945449095534 T^{10} - 16894314638903 T^{11} + 185758185266123 T^{12} - 16894314638903 p T^{13} + 1945449095534 p^{2} T^{14} - 151747795747 p^{3} T^{15} + 15277660667 p^{4} T^{16} - 996405958 p^{5} T^{17} + 87697456 p^{6} T^{18} - 4560335 p^{7} T^{19} + 348984 p^{8} T^{20} - 13116 p^{9} T^{21} + 863 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 - 24 T + 578 T^{2} - 8123 T^{3} + 131247 T^{4} - 1610393 T^{5} + 23145950 T^{6} - 257958908 T^{7} + 3198014560 T^{8} - 31793265584 T^{9} + 360076976807 T^{10} - 3357276169589 T^{11} + 35144002539569 T^{12} - 3357276169589 p T^{13} + 360076976807 p^{2} T^{14} - 31793265584 p^{3} T^{15} + 3198014560 p^{4} T^{16} - 257958908 p^{5} T^{17} + 23145950 p^{6} T^{18} - 1610393 p^{7} T^{19} + 131247 p^{8} T^{20} - 8123 p^{9} T^{21} + 578 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 + 25 T + 1103 T^{2} + 21705 T^{3} + 551572 T^{4} + 8957656 T^{5} + 167995324 T^{6} + 2313751659 T^{7} + 34991740282 T^{8} + 414911018767 T^{9} + 5273346067898 T^{10} + 54184093348156 T^{11} + 590821306410541 T^{12} + 54184093348156 p T^{13} + 5273346067898 p^{2} T^{14} + 414911018767 p^{3} T^{15} + 34991740282 p^{4} T^{16} + 2313751659 p^{5} T^{17} + 167995324 p^{6} T^{18} + 8957656 p^{7} T^{19} + 551572 p^{8} T^{20} + 21705 p^{9} T^{21} + 1103 p^{10} T^{22} + 25 p^{11} T^{23} + p^{12} T^{24} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.23671850238021672710213780553, −1.88888537209609726301578358831, −1.87973679816920694651449836737, −1.76282584463544001255483445161, −1.75264059172404144745645352301, −1.73373678064987525836856913447, −1.72052007214366446129820805432, −1.71345715946999462378600293517, −1.60087766238926537706458266803, −1.58752221885432939926708736138, −1.49174672587043813329586672727, −1.48481388395072326007553031886, −1.24340086217206695307026680668, −1.20847776283533113375550546205, −0.889519521614863996493417070378, −0.879172602861975477086570577663, −0.833309187124058500429288039362, −0.791413120403158780761639582602, −0.69125529516605066899667742817, −0.59233792884023619034504634310, −0.51864563540575480112053900073, −0.45838413417870085481505307293, −0.38996306865383215032429799444, −0.33800035764872442640047898198, −0.22431865172392479639120307821, 0.22431865172392479639120307821, 0.33800035764872442640047898198, 0.38996306865383215032429799444, 0.45838413417870085481505307293, 0.51864563540575480112053900073, 0.59233792884023619034504634310, 0.69125529516605066899667742817, 0.791413120403158780761639582602, 0.833309187124058500429288039362, 0.879172602861975477086570577663, 0.889519521614863996493417070378, 1.20847776283533113375550546205, 1.24340086217206695307026680668, 1.48481388395072326007553031886, 1.49174672587043813329586672727, 1.58752221885432939926708736138, 1.60087766238926537706458266803, 1.71345715946999462378600293517, 1.72052007214366446129820805432, 1.73373678064987525836856913447, 1.75264059172404144745645352301, 1.76282584463544001255483445161, 1.87973679816920694651449836737, 1.88888537209609726301578358831, 2.23671850238021672710213780553

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

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