Properties

Degree 24
Conductor $ 2^{12} \cdot 3^{12} \cdot 13^{12} \cdot 103^{12} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 12

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·2-s + 12·3-s + 78·4-s − 4·5-s − 144·6-s − 364·8-s + 78·9-s + 48·10-s − 9·11-s + 936·12-s + 12·13-s − 48·15-s + 1.36e3·16-s − 20·17-s − 936·18-s + 4·19-s − 312·20-s + 108·22-s − 30·23-s − 4.36e3·24-s − 15·25-s − 144·26-s + 364·27-s − 29·29-s + 576·30-s + 6·31-s − 4.36e3·32-s + ⋯
L(s)  = 1  − 8.48·2-s + 6.92·3-s + 39·4-s − 1.78·5-s − 58.7·6-s − 128.·8-s + 26·9-s + 15.1·10-s − 2.71·11-s + 270.·12-s + 3.32·13-s − 12.3·15-s + 341.·16-s − 4.85·17-s − 220.·18-s + 0.917·19-s − 69.7·20-s + 23.0·22-s − 6.25·23-s − 891.·24-s − 3·25-s − 28.2·26-s + 70.0·27-s − 5.38·29-s + 105.·30-s + 1.07·31-s − 772.·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 13^{12} \cdot 103^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 13^{12} \cdot 103^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{12} \cdot 3^{12} \cdot 13^{12} \cdot 103^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8034} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(12\)
Selberg data  =  \((24,\ 2^{12} \cdot 3^{12} \cdot 13^{12} \cdot 103^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;13,\;103\}$,\(F_p(T)\) is a polynomial of degree 24. If $p \in \{2,\;3,\;13,\;103\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad2 \( ( 1 + T )^{12} \)
3 \( ( 1 - T )^{12} \)
13 \( ( 1 - T )^{12} \)
103 \( ( 1 + T )^{12} \)
good5 \( 1 + 4 T + 31 T^{2} + 93 T^{3} + 87 p T^{4} + 208 p T^{5} + 738 p T^{6} + 6832 T^{7} + 20687 T^{8} + 27902 T^{9} + 85823 T^{10} + 82249 T^{11} + 361034 T^{12} + 82249 p T^{13} + 85823 p^{2} T^{14} + 27902 p^{3} T^{15} + 20687 p^{4} T^{16} + 6832 p^{5} T^{17} + 738 p^{7} T^{18} + 208 p^{8} T^{19} + 87 p^{9} T^{20} + 93 p^{9} T^{21} + 31 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \)
7 \( 1 + 37 T^{2} + 6 T^{3} + 705 T^{4} + 234 T^{5} + 1310 p T^{6} + 4703 T^{7} + 91608 T^{8} + 62424 T^{9} + 762310 T^{10} + 591697 T^{11} + 5608434 T^{12} + 591697 p T^{13} + 762310 p^{2} T^{14} + 62424 p^{3} T^{15} + 91608 p^{4} T^{16} + 4703 p^{5} T^{17} + 1310 p^{7} T^{18} + 234 p^{7} T^{19} + 705 p^{8} T^{20} + 6 p^{9} T^{21} + 37 p^{10} T^{22} + p^{12} T^{24} \)
11 \( 1 + 9 T + 108 T^{2} + 662 T^{3} + 4825 T^{4} + 23523 T^{5} + 133338 T^{6} + 556308 T^{7} + 2669196 T^{8} + 892578 p T^{9} + 41239643 T^{10} + 135341926 T^{11} + 506238982 T^{12} + 135341926 p T^{13} + 41239643 p^{2} T^{14} + 892578 p^{4} T^{15} + 2669196 p^{4} T^{16} + 556308 p^{5} T^{17} + 133338 p^{6} T^{18} + 23523 p^{7} T^{19} + 4825 p^{8} T^{20} + 662 p^{9} T^{21} + 108 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 + 20 T + 308 T^{2} + 3239 T^{3} + 29136 T^{4} + 213080 T^{5} + 1395897 T^{6} + 7911158 T^{7} + 41497074 T^{8} + 11521088 p T^{9} + 887388135 T^{10} + 3759932727 T^{11} + 15860100946 T^{12} + 3759932727 p T^{13} + 887388135 p^{2} T^{14} + 11521088 p^{4} T^{15} + 41497074 p^{4} T^{16} + 7911158 p^{5} T^{17} + 1395897 p^{6} T^{18} + 213080 p^{7} T^{19} + 29136 p^{8} T^{20} + 3239 p^{9} T^{21} + 308 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 - 4 T + 126 T^{2} - 356 T^{3} + 369 p T^{4} - 13474 T^{5} + 244168 T^{6} - 330157 T^{7} + 6604515 T^{8} - 7926739 T^{9} + 156854122 T^{10} - 194231596 T^{11} + 3243988882 T^{12} - 194231596 p T^{13} + 156854122 p^{2} T^{14} - 7926739 p^{3} T^{15} + 6604515 p^{4} T^{16} - 330157 p^{5} T^{17} + 244168 p^{6} T^{18} - 13474 p^{7} T^{19} + 369 p^{9} T^{20} - 356 p^{9} T^{21} + 126 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 + 30 T + 568 T^{2} + 7949 T^{3} + 90946 T^{4} + 886648 T^{5} + 7594755 T^{6} + 58167924 T^{7} + 403735953 T^{8} + 2561028471 T^{9} + 14941629527 T^{10} + 80475187848 T^{11} + 401092920340 T^{12} + 80475187848 p T^{13} + 14941629527 p^{2} T^{14} + 2561028471 p^{3} T^{15} + 403735953 p^{4} T^{16} + 58167924 p^{5} T^{17} + 7594755 p^{6} T^{18} + 886648 p^{7} T^{19} + 90946 p^{8} T^{20} + 7949 p^{9} T^{21} + 568 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 + p T + 581 T^{2} + 8215 T^{3} + 96385 T^{4} + 943160 T^{5} + 8220530 T^{6} + 63975721 T^{7} + 462363625 T^{8} + 3089568988 T^{9} + 19514878297 T^{10} + 115038001803 T^{11} + 641439793194 T^{12} + 115038001803 p T^{13} + 19514878297 p^{2} T^{14} + 3089568988 p^{3} T^{15} + 462363625 p^{4} T^{16} + 63975721 p^{5} T^{17} + 8220530 p^{6} T^{18} + 943160 p^{7} T^{19} + 96385 p^{8} T^{20} + 8215 p^{9} T^{21} + 581 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \)
31 \( 1 - 6 T + 202 T^{2} - 1066 T^{3} + 18815 T^{4} - 88824 T^{5} + 1079326 T^{6} - 4570047 T^{7} + 43249037 T^{8} - 166059267 T^{9} + 1369526848 T^{10} - 5039552536 T^{11} + 41133646998 T^{12} - 5039552536 p T^{13} + 1369526848 p^{2} T^{14} - 166059267 p^{3} T^{15} + 43249037 p^{4} T^{16} - 4570047 p^{5} T^{17} + 1079326 p^{6} T^{18} - 88824 p^{7} T^{19} + 18815 p^{8} T^{20} - 1066 p^{9} T^{21} + 202 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 - 7 T + 266 T^{2} - 1606 T^{3} + 35104 T^{4} - 190635 T^{5} + 3077220 T^{6} - 15279486 T^{7} + 200225468 T^{8} - 912382020 T^{9} + 10212857752 T^{10} - 42495240726 T^{11} + 419463956610 T^{12} - 42495240726 p T^{13} + 10212857752 p^{2} T^{14} - 912382020 p^{3} T^{15} + 200225468 p^{4} T^{16} - 15279486 p^{5} T^{17} + 3077220 p^{6} T^{18} - 190635 p^{7} T^{19} + 35104 p^{8} T^{20} - 1606 p^{9} T^{21} + 266 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 + 8 T + 210 T^{2} + 1836 T^{3} + 25968 T^{4} + 214790 T^{5} + 2323316 T^{6} + 17480343 T^{7} + 160141110 T^{8} + 1094530654 T^{9} + 8822694258 T^{10} + 55173358345 T^{11} + 397594665394 T^{12} + 55173358345 p T^{13} + 8822694258 p^{2} T^{14} + 1094530654 p^{3} T^{15} + 160141110 p^{4} T^{16} + 17480343 p^{5} T^{17} + 2323316 p^{6} T^{18} + 214790 p^{7} T^{19} + 25968 p^{8} T^{20} + 1836 p^{9} T^{21} + 210 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 + 8 T + 243 T^{2} + 1570 T^{3} + 28840 T^{4} + 150146 T^{5} + 2269784 T^{6} + 9907625 T^{7} + 140394002 T^{8} + 545817614 T^{9} + 7474610385 T^{10} + 26919387673 T^{11} + 346179346594 T^{12} + 26919387673 p T^{13} + 7474610385 p^{2} T^{14} + 545817614 p^{3} T^{15} + 140394002 p^{4} T^{16} + 9907625 p^{5} T^{17} + 2269784 p^{6} T^{18} + 150146 p^{7} T^{19} + 28840 p^{8} T^{20} + 1570 p^{9} T^{21} + 243 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 + 16 T + 357 T^{2} + 3968 T^{3} + 54911 T^{4} + 497508 T^{5} + 5443282 T^{6} + 42837179 T^{7} + 403460171 T^{8} + 2847775608 T^{9} + 24127276587 T^{10} + 156134470657 T^{11} + 1220882385238 T^{12} + 156134470657 p T^{13} + 24127276587 p^{2} T^{14} + 2847775608 p^{3} T^{15} + 403460171 p^{4} T^{16} + 42837179 p^{5} T^{17} + 5443282 p^{6} T^{18} + 497508 p^{7} T^{19} + 54911 p^{8} T^{20} + 3968 p^{9} T^{21} + 357 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 + 9 T + 364 T^{2} + 2648 T^{3} + 63946 T^{4} + 380636 T^{5} + 7277843 T^{6} + 35772976 T^{7} + 611076458 T^{8} + 2519222669 T^{9} + 41087609373 T^{10} + 147982600782 T^{11} + 2340259069438 T^{12} + 147982600782 p T^{13} + 41087609373 p^{2} T^{14} + 2519222669 p^{3} T^{15} + 611076458 p^{4} T^{16} + 35772976 p^{5} T^{17} + 7277843 p^{6} T^{18} + 380636 p^{7} T^{19} + 63946 p^{8} T^{20} + 2648 p^{9} T^{21} + 364 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 + 29 T + 751 T^{2} + 234 p T^{3} + 226698 T^{4} + 3155964 T^{5} + 40064037 T^{6} + 456699304 T^{7} + 4815443937 T^{8} + 46780459281 T^{9} + 424380475788 T^{10} + 3590840392348 T^{11} + 28503939680408 T^{12} + 3590840392348 p T^{13} + 424380475788 p^{2} T^{14} + 46780459281 p^{3} T^{15} + 4815443937 p^{4} T^{16} + 456699304 p^{5} T^{17} + 40064037 p^{6} T^{18} + 3155964 p^{7} T^{19} + 226698 p^{8} T^{20} + 234 p^{10} T^{21} + 751 p^{10} T^{22} + 29 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 + 26 T + 811 T^{2} + 15010 T^{3} + 278860 T^{4} + 4057732 T^{5} + 56493224 T^{6} + 678244599 T^{7} + 7664308564 T^{8} + 77893196608 T^{9} + 740665052183 T^{10} + 6454026745289 T^{11} + 52513812625194 T^{12} + 6454026745289 p T^{13} + 740665052183 p^{2} T^{14} + 77893196608 p^{3} T^{15} + 7664308564 p^{4} T^{16} + 678244599 p^{5} T^{17} + 56493224 p^{6} T^{18} + 4057732 p^{7} T^{19} + 278860 p^{8} T^{20} + 15010 p^{9} T^{21} + 811 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 - 12 T + 541 T^{2} - 6152 T^{3} + 146173 T^{4} - 1560061 T^{5} + 25832558 T^{6} - 255792465 T^{7} + 3306058168 T^{8} - 29991143507 T^{9} + 321676689949 T^{10} - 2633052193083 T^{11} + 24350241505332 T^{12} - 2633052193083 p T^{13} + 321676689949 p^{2} T^{14} - 29991143507 p^{3} T^{15} + 3306058168 p^{4} T^{16} - 255792465 p^{5} T^{17} + 25832558 p^{6} T^{18} - 1560061 p^{7} T^{19} + 146173 p^{8} T^{20} - 6152 p^{9} T^{21} + 541 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 + 35 T + 1135 T^{2} + 24696 T^{3} + 491537 T^{4} + 8021287 T^{5} + 120877529 T^{6} + 1593235761 T^{7} + 19548155567 T^{8} + 215684766884 T^{9} + 2226478765392 T^{10} + 20926580550085 T^{11} + 184460124370606 T^{12} + 20926580550085 p T^{13} + 2226478765392 p^{2} T^{14} + 215684766884 p^{3} T^{15} + 19548155567 p^{4} T^{16} + 1593235761 p^{5} T^{17} + 120877529 p^{6} T^{18} + 8021287 p^{7} T^{19} + 491537 p^{8} T^{20} + 24696 p^{9} T^{21} + 1135 p^{10} T^{22} + 35 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 - 18 T + 354 T^{2} - 5804 T^{3} + 82357 T^{4} - 1064849 T^{5} + 13002100 T^{6} - 144999477 T^{7} + 1559317972 T^{8} - 15575597371 T^{9} + 2063675014 p T^{10} - 1373951341057 T^{11} + 12006180010132 T^{12} - 1373951341057 p T^{13} + 2063675014 p^{3} T^{14} - 15575597371 p^{3} T^{15} + 1559317972 p^{4} T^{16} - 144999477 p^{5} T^{17} + 13002100 p^{6} T^{18} - 1064849 p^{7} T^{19} + 82357 p^{8} T^{20} - 5804 p^{9} T^{21} + 354 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 + 37 T + 1105 T^{2} + 25037 T^{3} + 491127 T^{4} + 8319158 T^{5} + 127058602 T^{6} + 1747643419 T^{7} + 22062467511 T^{8} + 255455090796 T^{9} + 2741158508173 T^{10} + 27197618826911 T^{11} + 251063744352914 T^{12} + 27197618826911 p T^{13} + 2741158508173 p^{2} T^{14} + 255455090796 p^{3} T^{15} + 22062467511 p^{4} T^{16} + 1747643419 p^{5} T^{17} + 127058602 p^{6} T^{18} + 8319158 p^{7} T^{19} + 491127 p^{8} T^{20} + 25037 p^{9} T^{21} + 1105 p^{10} T^{22} + 37 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 + 24 T + 753 T^{2} + 12791 T^{3} + 240301 T^{4} + 3213851 T^{5} + 45719477 T^{6} + 510509135 T^{7} + 6050773221 T^{8} + 59078147485 T^{9} + 622265694970 T^{10} + 5563156908874 T^{11} + 54682684150922 T^{12} + 5563156908874 p T^{13} + 622265694970 p^{2} T^{14} + 59078147485 p^{3} T^{15} + 6050773221 p^{4} T^{16} + 510509135 p^{5} T^{17} + 45719477 p^{6} T^{18} + 3213851 p^{7} T^{19} + 240301 p^{8} T^{20} + 12791 p^{9} T^{21} + 753 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 - 15 T + 841 T^{2} - 11662 T^{3} + 344802 T^{4} - 4319083 T^{5} + 89943638 T^{6} - 1006107027 T^{7} + 16494253885 T^{8} - 163662185312 T^{9} + 2230819592745 T^{10} - 19500674759813 T^{11} + 227730176149968 T^{12} - 19500674759813 p T^{13} + 2230819592745 p^{2} T^{14} - 163662185312 p^{3} T^{15} + 16494253885 p^{4} T^{16} - 1006107027 p^{5} T^{17} + 89943638 p^{6} T^{18} - 4319083 p^{7} T^{19} + 344802 p^{8} T^{20} - 11662 p^{9} T^{21} + 841 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 + 11 T + 1026 T^{2} + 10059 T^{3} + 492409 T^{4} + 4310715 T^{5} + 146403265 T^{6} + 1142816468 T^{7} + 30068791037 T^{8} + 208306392171 T^{9} + 4498037411981 T^{10} + 27417640536048 T^{11} + 502791327262770 T^{12} + 27417640536048 p T^{13} + 4498037411981 p^{2} T^{14} + 208306392171 p^{3} T^{15} + 30068791037 p^{4} T^{16} + 1142816468 p^{5} T^{17} + 146403265 p^{6} T^{18} + 4310715 p^{7} T^{19} + 492409 p^{8} T^{20} + 10059 p^{9} T^{21} + 1026 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \)
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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

−2.57855226391466602717450463828, −2.43318935538383437149290873078, −2.38321975200674647942478408972, −2.34820819707897882151123709867, −2.26367317398871337325454746350, −2.16069953989833725867600471252, −2.14133460528550893024401579179, −2.13435392783651159413947587409, −2.05185962892504425154692456490, −2.03298156282634615004203543434, −2.02802309764199751788894174472, −1.95233650972284143874528992049, −1.82916658007768075344499662719, −1.74988949318623385051823895131, −1.60374082069831512361915448633, −1.49447603746591919225882586726, −1.36332269886466537191373427638, −1.32221705223917583617736199467, −1.29585754440563879879002959740, −1.25124801471604180707492844182, −1.24753528825188337223263338883, −1.22579928017026652305299717433, −1.21670443082506356069760358187, −1.21463325897511213215816626106, −0.915475189864899244638395204044, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.915475189864899244638395204044, 1.21463325897511213215816626106, 1.21670443082506356069760358187, 1.22579928017026652305299717433, 1.24753528825188337223263338883, 1.25124801471604180707492844182, 1.29585754440563879879002959740, 1.32221705223917583617736199467, 1.36332269886466537191373427638, 1.49447603746591919225882586726, 1.60374082069831512361915448633, 1.74988949318623385051823895131, 1.82916658007768075344499662719, 1.95233650972284143874528992049, 2.02802309764199751788894174472, 2.03298156282634615004203543434, 2.05185962892504425154692456490, 2.13435392783651159413947587409, 2.14133460528550893024401579179, 2.16069953989833725867600471252, 2.26367317398871337325454746350, 2.34820819707897882151123709867, 2.38321975200674647942478408972, 2.43318935538383437149290873078, 2.57855226391466602717450463828

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

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