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Properties

Label 12-2e36-1.1-c16e6-0-0
Degree $12$
Conductor $68719476736$
Sign $1$
Analytic cond. $1.25714\times 10^{12}$
Root an. cond. $10.1925$
Motivic weight $16$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.06e5·5-s + 6.03e7·9-s − 2.54e9·13-s + 1.57e9·17-s − 1.93e11·25-s + 1.15e12·29-s − 8.58e12·37-s + 1.84e12·41-s + 3.05e13·45-s + 9.69e13·49-s − 1.30e14·53-s + 4.29e14·61-s − 1.28e15·65-s − 1.14e15·73-s + 2.71e14·81-s + 8.00e14·85-s + 2.04e16·89-s + 4.18e15·97-s + 3.77e16·101-s + 6.16e16·109-s + 7.41e16·113-s − 1.53e17·117-s + 2.37e17·121-s − 6.30e16·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.29·5-s + 1.40·9-s − 3.11·13-s + 0.226·17-s − 1.26·25-s + 2.31·29-s − 2.44·37-s + 0.230·41-s + 1.81·45-s + 2.91·49-s − 2.09·53-s + 2.24·61-s − 4.04·65-s − 1.42·73-s + 0.146·81-s + 0.293·85-s + 5.18·89-s + 0.533·97-s + 3.48·101-s + 3.09·109-s + 2.79·113-s − 4.37·117-s + 5.16·121-s − 1.05·125-s + 3.00·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(17-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+8)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{36}\)
Sign: $1$
Analytic conductor: \(1.25714\times 10^{12}\)
Root analytic conductor: \(10.1925\)
Motivic weight: \(16\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{36} ,\ ( \ : [8]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.2949696529\)
\(L(\frac12)\) \(\approx\) \(0.2949696529\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 6705878 p^{2} T^{2} + 13873090274293 p^{5} T^{4} - 2675500055803584244 p^{10} T^{6} + 13873090274293 p^{37} T^{8} - 6705878 p^{66} T^{10} + p^{96} T^{12} \)
5 \( ( 1 - 50674 p T + 1544991163 p^{3} T^{2} - 132367498504748 p^{4} T^{3} + 1544991163 p^{19} T^{4} - 50674 p^{33} T^{5} + p^{48} T^{6} )^{2} \)
7 \( 1 - 13847881261386 p T^{2} + \)\(15\!\cdots\!93\)\( p^{3} T^{4} - \)\(26\!\cdots\!72\)\( p^{7} T^{6} + \)\(15\!\cdots\!93\)\( p^{35} T^{8} - 13847881261386 p^{65} T^{10} + p^{96} T^{12} \)
11 \( 1 - 237379368948178566 T^{2} + \)\(18\!\cdots\!05\)\( p^{3} T^{4} - \)\(10\!\cdots\!00\)\( p^{4} T^{6} + \)\(18\!\cdots\!05\)\( p^{35} T^{8} - 237379368948178566 p^{64} T^{10} + p^{96} T^{12} \)
13 \( ( 1 + 97864542 p T + 12179343757966263 p^{2} T^{2} + \)\(71\!\cdots\!16\)\( p^{3} T^{3} + 12179343757966263 p^{18} T^{4} + 97864542 p^{33} T^{5} + p^{48} T^{6} )^{2} \)
17 \( ( 1 - 789602566 T + 57000831538845708047 T^{2} - \)\(28\!\cdots\!32\)\( T^{3} + 57000831538845708047 p^{16} T^{4} - 789602566 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
19 \( 1 - \)\(43\!\cdots\!26\)\( T^{2} + \)\(95\!\cdots\!15\)\( T^{4} - \)\(20\!\cdots\!40\)\( T^{6} + \)\(95\!\cdots\!15\)\( p^{32} T^{8} - \)\(43\!\cdots\!26\)\( p^{64} T^{10} + p^{96} T^{12} \)
23 \( 1 - \)\(32\!\cdots\!02\)\( T^{2} + \)\(45\!\cdots\!19\)\( T^{4} - \)\(36\!\cdots\!36\)\( T^{6} + \)\(45\!\cdots\!19\)\( p^{32} T^{8} - \)\(32\!\cdots\!02\)\( p^{64} T^{10} + p^{96} T^{12} \)
29 \( ( 1 - 579205884218 T + \)\(71\!\cdots\!39\)\( T^{2} - \)\(24\!\cdots\!64\)\( T^{3} + \)\(71\!\cdots\!39\)\( p^{16} T^{4} - 579205884218 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
31 \( 1 - \)\(20\!\cdots\!26\)\( T^{2} + \)\(14\!\cdots\!15\)\( T^{4} - \)\(76\!\cdots\!40\)\( p^{2} T^{6} + \)\(14\!\cdots\!15\)\( p^{32} T^{8} - \)\(20\!\cdots\!26\)\( p^{64} T^{10} + p^{96} T^{12} \)
37 \( ( 1 + 4290723009606 T + \)\(41\!\cdots\!47\)\( T^{2} + \)\(10\!\cdots\!12\)\( T^{3} + \)\(41\!\cdots\!47\)\( p^{16} T^{4} + 4290723009606 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
41 \( ( 1 - 920184626566 T + \)\(13\!\cdots\!35\)\( T^{2} + \)\(75\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!35\)\( p^{16} T^{4} - 920184626566 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
43 \( 1 - \)\(28\!\cdots\!02\)\( T^{2} + \)\(63\!\cdots\!99\)\( T^{4} - \)\(10\!\cdots\!96\)\( T^{6} + \)\(63\!\cdots\!99\)\( p^{32} T^{8} - \)\(28\!\cdots\!02\)\( p^{64} T^{10} + p^{96} T^{12} \)
47 \( 1 - \)\(13\!\cdots\!42\)\( T^{2} + \)\(11\!\cdots\!99\)\( T^{4} - \)\(68\!\cdots\!76\)\( T^{6} + \)\(11\!\cdots\!99\)\( p^{32} T^{8} - \)\(13\!\cdots\!42\)\( p^{64} T^{10} + p^{96} T^{12} \)
53 \( ( 1 + 65334134704966 T + \)\(78\!\cdots\!07\)\( T^{2} + \)\(36\!\cdots\!12\)\( T^{3} + \)\(78\!\cdots\!07\)\( p^{16} T^{4} + 65334134704966 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
59 \( 1 - \)\(10\!\cdots\!66\)\( T^{2} + \)\(47\!\cdots\!35\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(47\!\cdots\!35\)\( p^{32} T^{8} - \)\(10\!\cdots\!66\)\( p^{64} T^{10} + p^{96} T^{12} \)
61 \( ( 1 - 214743004157754 T + \)\(58\!\cdots\!15\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!15\)\( p^{16} T^{4} - 214743004157754 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
67 \( 1 - \)\(81\!\cdots\!22\)\( T^{2} + \)\(29\!\cdots\!79\)\( T^{4} - \)\(63\!\cdots\!36\)\( T^{6} + \)\(29\!\cdots\!79\)\( p^{32} T^{8} - \)\(81\!\cdots\!22\)\( p^{64} T^{10} + p^{96} T^{12} \)
71 \( 1 - \)\(20\!\cdots\!86\)\( T^{2} + \)\(53\!\cdots\!95\)\( T^{4} - \)\(73\!\cdots\!60\)\( T^{6} + \)\(53\!\cdots\!95\)\( p^{32} T^{8} - \)\(20\!\cdots\!86\)\( p^{64} T^{10} + p^{96} T^{12} \)
73 \( ( 1 + 572800519885434 T + \)\(19\!\cdots\!47\)\( T^{2} + \)\(70\!\cdots\!28\)\( T^{3} + \)\(19\!\cdots\!47\)\( p^{16} T^{4} + 572800519885434 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
79 \( 1 - \)\(57\!\cdots\!66\)\( T^{2} + \)\(23\!\cdots\!15\)\( T^{4} - \)\(59\!\cdots\!40\)\( T^{6} + \)\(23\!\cdots\!15\)\( p^{32} T^{8} - \)\(57\!\cdots\!66\)\( p^{64} T^{10} + p^{96} T^{12} \)
83 \( 1 - \)\(70\!\cdots\!42\)\( T^{2} + \)\(93\!\cdots\!79\)\( T^{4} - \)\(37\!\cdots\!96\)\( T^{6} + \)\(93\!\cdots\!79\)\( p^{32} T^{8} - \)\(70\!\cdots\!42\)\( p^{64} T^{10} + p^{96} T^{12} \)
89 \( ( 1 - 10213379497545862 T + \)\(80\!\cdots\!39\)\( T^{2} - \)\(35\!\cdots\!36\)\( T^{3} + \)\(80\!\cdots\!39\)\( p^{16} T^{4} - 10213379497545862 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
97 \( ( 1 - 2091198277201926 T + \)\(17\!\cdots\!87\)\( T^{2} - \)\(24\!\cdots\!52\)\( T^{3} + \)\(17\!\cdots\!87\)\( p^{16} T^{4} - 2091198277201926 p^{32} T^{5} + p^{48} T^{6} )^{2} \)
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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

−5.43423245406895999519773924698, −5.05931771023213569211823162785, −4.75850099169081720396694155531, −4.68048581092799760593938399676, −4.65048946868911844211323642906, −4.46367729688238093154556507568, −4.42798179991451847506207104100, −3.74129156094020107980366413368, −3.51046140028253551880142339381, −3.50984360075372047568100391749, −3.28471087584462781430301684455, −3.03199339730115052305022855858, −2.73869442600058249382251405274, −2.36819310917881637572316485579, −2.17166736023070128345529133542, −2.07178578602981407572993957603, −1.95399409862987628003704839663, −1.92324220781150209091729437596, −1.70188145466167556440732561058, −1.16899262279250926003411825765, −0.927216199697044053014307205376, −0.72766129757657222199347357515, −0.61079088783607264446611136672, −0.54554930894547731887683365950, −0.02750859851146188125128685994, 0.02750859851146188125128685994, 0.54554930894547731887683365950, 0.61079088783607264446611136672, 0.72766129757657222199347357515, 0.927216199697044053014307205376, 1.16899262279250926003411825765, 1.70188145466167556440732561058, 1.92324220781150209091729437596, 1.95399409862987628003704839663, 2.07178578602981407572993957603, 2.17166736023070128345529133542, 2.36819310917881637572316485579, 2.73869442600058249382251405274, 3.03199339730115052305022855858, 3.28471087584462781430301684455, 3.50984360075372047568100391749, 3.51046140028253551880142339381, 3.74129156094020107980366413368, 4.42798179991451847506207104100, 4.46367729688238093154556507568, 4.65048946868911844211323642906, 4.68048581092799760593938399676, 4.75850099169081720396694155531, 5.05931771023213569211823162785, 5.43423245406895999519773924698

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

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