Properties

Degree $12$
Conductor $15625$
Sign $1$
Motivic weight $8$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 72·3-s + 2·4-s + 220·5-s + 144·6-s − 2.35e3·7-s − 2.91e3·8-s + 2.59e3·9-s − 440·10-s + 2.31e4·11-s − 144·12-s − 1.19e5·13-s + 4.70e3·14-s − 1.58e4·15-s + 6.27e4·16-s − 2.65e5·17-s − 5.18e3·18-s + 440·20-s + 1.69e5·21-s − 4.63e4·22-s + 2.88e4·23-s + 2.09e5·24-s − 1.45e5·25-s + 2.38e5·26-s − 8.89e4·27-s − 4.70e3·28-s + 3.16e4·30-s + ⋯
L(s)  = 1  − 1/8·2-s − 8/9·3-s + 0.00781·4-s + 0.351·5-s + 1/9·6-s − 0.979·7-s − 0.710·8-s + 0.395·9-s − 0.0439·10-s + 1.58·11-s − 0.00694·12-s − 4.17·13-s + 6/49·14-s − 0.312·15-s + 0.957·16-s − 3.17·17-s − 0.0493·18-s + 0.00274·20-s + 0.870·21-s − 0.198·22-s + 0.103·23-s + 0.631·24-s − 0.373·25-s + 0.521·26-s − 0.167·27-s − 0.00765·28-s + 0.0391·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s+4)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(15625\)    =    \(5^{6}\)
Sign: $1$
Motivic weight: \(8\)
Character: induced by $\chi_{5} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 15625,\ (\ :[4]^{6}),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0185895\)
\(L(\frac12)\) \(\approx\) \(0.0185895\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 44 p T + 311 p^{4} T^{2} - 2552 p^{7} T^{3} + 311 p^{12} T^{4} - 44 p^{17} T^{5} + p^{24} T^{6} \)
good2 \( 1 + p T + p T^{2} + 91 p^{5} T^{3} - 799 p^{6} T^{4} - 8873 p^{7} T^{5} + 16177 p^{7} T^{6} - 8873 p^{15} T^{7} - 799 p^{22} T^{8} + 91 p^{29} T^{9} + p^{33} T^{10} + p^{41} T^{11} + p^{48} T^{12} \)
3 \( 1 + 8 p^{2} T + 32 p^{4} T^{2} + 3296 p^{3} T^{3} - 4782569 p^{2} T^{4} - 45184904 p^{4} T^{5} - 203005216 p^{6} T^{6} - 45184904 p^{12} T^{7} - 4782569 p^{18} T^{8} + 3296 p^{27} T^{9} + 32 p^{36} T^{10} + 8 p^{42} T^{11} + p^{48} T^{12} \)
7 \( 1 + 48 p^{2} T + 1152 p^{4} T^{2} - 6884936 p^{3} T^{3} + 2619602799 p^{4} T^{4} + 7645236875928 p^{5} T^{5} + 2444629423375904 p^{6} T^{6} + 7645236875928 p^{13} T^{7} + 2619602799 p^{20} T^{8} - 6884936 p^{27} T^{9} + 1152 p^{36} T^{10} + 48 p^{42} T^{11} + p^{48} T^{12} \)
11 \( ( 1 - 11596 T + 493098215 T^{2} - 5106545516920 T^{3} + 493098215 p^{8} T^{4} - 11596 p^{16} T^{5} + p^{24} T^{6} )^{2} \)
13 \( 1 + 119142 T + 7097408082 T^{2} + 332686420223782 T^{3} + 13680551086291514559 T^{4} + \)\(46\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!76\)\( T^{6} + \)\(46\!\cdots\!56\)\( p^{8} T^{7} + 13680551086291514559 p^{16} T^{8} + 332686420223782 p^{24} T^{9} + 7097408082 p^{32} T^{10} + 119142 p^{40} T^{11} + p^{48} T^{12} \)
17 \( 1 + 265502 T + 35245656002 T^{2} + 3505767378301982 T^{3} + \)\(37\!\cdots\!19\)\( T^{4} + \)\(40\!\cdots\!76\)\( T^{5} + \)\(37\!\cdots\!76\)\( T^{6} + \)\(40\!\cdots\!76\)\( p^{8} T^{7} + \)\(37\!\cdots\!19\)\( p^{16} T^{8} + 3505767378301982 p^{24} T^{9} + 35245656002 p^{32} T^{10} + 265502 p^{40} T^{11} + p^{48} T^{12} \)
19 \( 1 - 66391003446 T^{2} + \)\(21\!\cdots\!15\)\( T^{4} - \)\(42\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!15\)\( p^{16} T^{8} - 66391003446 p^{32} T^{10} + p^{48} T^{12} \)
23 \( 1 - 1256 p T + 788768 p^{2} T^{2} - 2207314762520128 T^{3} + \)\(86\!\cdots\!39\)\( T^{4} - \)\(83\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} - \)\(83\!\cdots\!64\)\( p^{8} T^{7} + \)\(86\!\cdots\!39\)\( p^{16} T^{8} - 2207314762520128 p^{24} T^{9} + 788768 p^{34} T^{10} - 1256 p^{41} T^{11} + p^{48} T^{12} \)
29 \( 1 - 1726260912966 T^{2} + \)\(13\!\cdots\!15\)\( T^{4} - \)\(77\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!15\)\( p^{16} T^{8} - 1726260912966 p^{32} T^{10} + p^{48} T^{12} \)
31 \( ( 1 + 373824 T + 1438821265815 T^{2} + 103488660558742480 T^{3} + 1438821265815 p^{8} T^{4} + 373824 p^{16} T^{5} + p^{24} T^{6} )^{2} \)
37 \( 1 + 454002 T + 103058908002 T^{2} + 1591257258997412242 T^{3} + \)\(36\!\cdots\!59\)\( T^{4} + \)\(11\!\cdots\!36\)\( T^{5} + \)\(25\!\cdots\!36\)\( T^{6} + \)\(11\!\cdots\!36\)\( p^{8} T^{7} + \)\(36\!\cdots\!59\)\( p^{16} T^{8} + 1591257258997412242 p^{24} T^{9} + 103058908002 p^{32} T^{10} + 454002 p^{40} T^{11} + p^{48} T^{12} \)
41 \( ( 1 - 1244716 T + 19779856453415 T^{2} - 17348876621060070520 T^{3} + 19779856453415 p^{8} T^{4} - 1244716 p^{16} T^{5} + p^{24} T^{6} )^{2} \)
43 \( 1 - 792648 T + 314145425952 T^{2} - 6701894073526462448 T^{3} + \)\(27\!\cdots\!99\)\( T^{4} - \)\(18\!\cdots\!04\)\( T^{5} + \)\(86\!\cdots\!96\)\( T^{6} - \)\(18\!\cdots\!04\)\( p^{8} T^{7} + \)\(27\!\cdots\!99\)\( p^{16} T^{8} - 6701894073526462448 p^{24} T^{9} + 314145425952 p^{32} T^{10} - 792648 p^{40} T^{11} + p^{48} T^{12} \)
47 \( 1 + 325816 p T + 53078032928 p^{2} T^{2} + \)\(85\!\cdots\!72\)\( T^{3} + \)\(63\!\cdots\!79\)\( T^{4} + \)\(35\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!16\)\( T^{6} + \)\(35\!\cdots\!16\)\( p^{8} T^{7} + \)\(63\!\cdots\!79\)\( p^{16} T^{8} + \)\(85\!\cdots\!72\)\( p^{24} T^{9} + 53078032928 p^{34} T^{10} + 325816 p^{41} T^{11} + p^{48} T^{12} \)
53 \( 1 + 13509122 T + 91248188605442 T^{2} + \)\(98\!\cdots\!42\)\( T^{3} + \)\(11\!\cdots\!79\)\( T^{4} + \)\(81\!\cdots\!76\)\( T^{5} + \)\(50\!\cdots\!36\)\( T^{6} + \)\(81\!\cdots\!76\)\( p^{8} T^{7} + \)\(11\!\cdots\!79\)\( p^{16} T^{8} + \)\(98\!\cdots\!42\)\( p^{24} T^{9} + 91248188605442 p^{32} T^{10} + 13509122 p^{40} T^{11} + p^{48} T^{12} \)
59 \( 1 - 413223229068726 T^{2} + \)\(98\!\cdots\!15\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(98\!\cdots\!15\)\( p^{16} T^{8} - 413223229068726 p^{32} T^{10} + p^{48} T^{12} \)
61 \( ( 1 - 12055596 T + 200152007609415 T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + 200152007609415 p^{8} T^{4} - 12055596 p^{16} T^{5} + p^{24} T^{6} )^{2} \)
67 \( 1 + 32827752 T + 538830650686752 T^{2} + \)\(16\!\cdots\!32\)\( T^{3} + \)\(54\!\cdots\!19\)\( T^{4} + \)\(88\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!76\)\( T^{6} + \)\(88\!\cdots\!76\)\( p^{8} T^{7} + \)\(54\!\cdots\!19\)\( p^{16} T^{8} + \)\(16\!\cdots\!32\)\( p^{24} T^{9} + 538830650686752 p^{32} T^{10} + 32827752 p^{40} T^{11} + p^{48} T^{12} \)
71 \( ( 1 + 6996464 T + 1071469029384215 T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + 1071469029384215 p^{8} T^{4} + 6996464 p^{16} T^{5} + p^{24} T^{6} )^{2} \)
73 \( 1 - 111859638 T + 6256289306745522 T^{2} - \)\(29\!\cdots\!78\)\( T^{3} + \)\(12\!\cdots\!39\)\( T^{4} - \)\(42\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} - \)\(42\!\cdots\!64\)\( p^{8} T^{7} + \)\(12\!\cdots\!39\)\( p^{16} T^{8} - \)\(29\!\cdots\!78\)\( p^{24} T^{9} + 6256289306745522 p^{32} T^{10} - 111859638 p^{40} T^{11} + p^{48} T^{12} \)
79 \( 1 - 4185433961698566 T^{2} + \)\(55\!\cdots\!15\)\( T^{4} - \)\(46\!\cdots\!20\)\( T^{6} + \)\(55\!\cdots\!15\)\( p^{16} T^{8} - 4185433961698566 p^{32} T^{10} + p^{48} T^{12} \)
83 \( 1 + 14768432 T + 109053291869312 T^{2} + \)\(28\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!19\)\( T^{4} + \)\(10\!\cdots\!16\)\( T^{5} + \)\(83\!\cdots\!56\)\( T^{6} + \)\(10\!\cdots\!16\)\( p^{8} T^{7} + \)\(10\!\cdots\!19\)\( p^{16} T^{8} + \)\(28\!\cdots\!12\)\( p^{24} T^{9} + 109053291869312 p^{32} T^{10} + 14768432 p^{40} T^{11} + p^{48} T^{12} \)
89 \( 1 - 8165894455313286 T^{2} + \)\(62\!\cdots\!15\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{6} + \)\(62\!\cdots\!15\)\( p^{16} T^{8} - 8165894455313286 p^{32} T^{10} + p^{48} T^{12} \)
97 \( 1 + 186656202 T + 17420268872532402 T^{2} + \)\(93\!\cdots\!22\)\( T^{3} + \)\(66\!\cdots\!79\)\( T^{4} + \)\(11\!\cdots\!16\)\( T^{5} + \)\(13\!\cdots\!16\)\( T^{6} + \)\(11\!\cdots\!16\)\( p^{8} T^{7} + \)\(66\!\cdots\!79\)\( p^{16} T^{8} + \)\(93\!\cdots\!22\)\( p^{24} T^{9} + 17420268872532402 p^{32} T^{10} + 186656202 p^{40} T^{11} + p^{48} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.07913462216372600186259149434, −12.76554466803163537011519474090, −12.40835243697841491853441164719, −12.06063191842044572364169484804, −11.91800541403415463872960774073, −11.49882662871319202047641892491, −11.09899152288409629868316510951, −10.80904127711720388284476460596, −10.03996946524320232548116239146, −9.927953871273442204192135892244, −9.481800254373481265993199787565, −9.258610739881257621293271378059, −9.148120791026819019623247058122, −8.153015598397907333233809721619, −7.82131647649432496149521622480, −6.90603413569605103844745153203, −6.80010731784266992744474803301, −6.56620737840845737433603586750, −5.92866625125254747864420880653, −4.99409213513494618218204955326, −4.84035359668290825174730410314, −3.97762592032845800453370154856, −2.76747802457835857559118822775, −2.09579531197545286769574704115, −0.06809824173264623395016738569, 0.06809824173264623395016738569, 2.09579531197545286769574704115, 2.76747802457835857559118822775, 3.97762592032845800453370154856, 4.84035359668290825174730410314, 4.99409213513494618218204955326, 5.92866625125254747864420880653, 6.56620737840845737433603586750, 6.80010731784266992744474803301, 6.90603413569605103844745153203, 7.82131647649432496149521622480, 8.153015598397907333233809721619, 9.148120791026819019623247058122, 9.258610739881257621293271378059, 9.481800254373481265993199787565, 9.927953871273442204192135892244, 10.03996946524320232548116239146, 10.80904127711720388284476460596, 11.09899152288409629868316510951, 11.49882662871319202047641892491, 11.91800541403415463872960774073, 12.06063191842044572364169484804, 12.40835243697841491853441164719, 12.76554466803163537011519474090, 13.07913462216372600186259149434

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

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