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Properties

Label 8-12e8-1.1-c12e4-0-4
Degree $8$
Conductor $429981696$
Sign $1$
Analytic cond. $3.00070\times 10^{8}$
Root an. cond. $11.4723$
Motivic weight $12$
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  + 2.19e4·5-s − 5.81e6·13-s + 3.38e7·17-s − 1.76e8·25-s − 4.25e7·29-s + 9.40e9·37-s − 1.79e10·41-s + 5.46e10·49-s − 6.05e10·53-s + 1.68e11·61-s − 1.27e11·65-s + 7.89e11·73-s + 7.44e11·85-s + 6.38e11·89-s − 5.63e11·97-s + 4.63e12·101-s − 5.39e12·109-s + 9.04e12·113-s + 4.12e12·121-s − 7.05e12·125-s + 127-s + 131-s + 137-s + 139-s − 9.35e11·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.40·5-s − 1.20·13-s + 1.40·17-s − 0.721·25-s − 0.0715·29-s + 3.66·37-s − 3.78·41-s + 3.94·49-s − 2.73·53-s + 3.26·61-s − 1.69·65-s + 5.21·73-s + 1.97·85-s + 1.28·89-s − 0.676·97-s + 4.37·101-s − 3.21·109-s + 4.34·113-s + 1.31·121-s − 1.85·125-s − 0.100·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(13-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+6)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(3.00070\times 10^{8}\)
Root analytic conductor: \(11.4723\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} ,\ ( \ : 6, 6, 6, 6 ),\ 1 )\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(15.52383466\)
\(L(\frac12)\) \(\approx\) \(15.52383466\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5$D_{4}$ \( ( 1 - 2196 p T + 430214 p^{4} T^{2} - 2196 p^{13} T^{3} + p^{24} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 54652997284 T^{2} + 23057278827917426598 p^{2} T^{4} - 54652997284 p^{24} T^{6} + p^{48} T^{8} \)
11$D_4\times C_2$ \( 1 - 34088557060 p^{2} T^{2} + \)\(16\!\cdots\!02\)\( p^{4} T^{4} - 34088557060 p^{26} T^{6} + p^{48} T^{8} \)
13$D_{4}$ \( ( 1 + 2905036 T - 1825385214714 T^{2} + 2905036 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 16941132 T + 1004522979764774 T^{2} - 16941132 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 5818731155199844 T^{2} + \)\(15\!\cdots\!02\)\( T^{4} - 5818731155199844 p^{24} T^{6} + p^{48} T^{8} \)
23$D_4\times C_2$ \( 1 - 40658686053902212 T^{2} + \)\(84\!\cdots\!22\)\( T^{4} - 40658686053902212 p^{24} T^{6} + p^{48} T^{8} \)
29$D_{4}$ \( ( 1 + 21291948 T + 565340476276432262 T^{2} + 21291948 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 1971519050733147940 T^{2} + \)\(19\!\cdots\!42\)\( T^{4} - 1971519050733147940 p^{24} T^{6} + p^{48} T^{8} \)
37$D_{4}$ \( ( 1 - 4701432868 T + 17305228534658117862 T^{2} - 4701432868 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 8984441268 T + 63939568982503208294 T^{2} + 8984441268 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 99624310717809132004 T^{2} + \)\(56\!\cdots\!82\)\( T^{4} - 99624310717809132004 p^{24} T^{6} + p^{48} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(10\!\cdots\!04\)\( T^{2} - \)\(11\!\cdots\!18\)\( T^{4} - \)\(10\!\cdots\!04\)\( p^{24} T^{6} + p^{48} T^{8} \)
53$D_{4}$ \( ( 1 + 30273478380 T + \)\(10\!\cdots\!06\)\( T^{2} + 30273478380 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - \)\(25\!\cdots\!60\)\( T^{2} + \)\(33\!\cdots\!22\)\( T^{4} - \)\(25\!\cdots\!60\)\( p^{24} T^{6} + p^{48} T^{8} \)
61$D_{4}$ \( ( 1 - 84143600836 T + \)\(70\!\cdots\!62\)\( T^{2} - 84143600836 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(49\!\cdots\!04\)\( T^{2} + \)\(27\!\cdots\!22\)\( T^{4} - \)\(49\!\cdots\!04\)\( p^{24} T^{6} + p^{48} T^{8} \)
71$D_4\times C_2$ \( 1 - \)\(63\!\cdots\!84\)\( T^{2} + \)\(15\!\cdots\!02\)\( T^{4} - \)\(63\!\cdots\!84\)\( p^{24} T^{6} + p^{48} T^{8} \)
73$D_{4}$ \( ( 1 - 394513314500 T + \)\(75\!\cdots\!46\)\( T^{2} - 394513314500 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!36\)\( T^{2} + \)\(12\!\cdots\!90\)\( T^{4} - \)\(14\!\cdots\!36\)\( p^{24} T^{6} + p^{48} T^{8} \)
83$D_4\times C_2$ \( 1 - \)\(37\!\cdots\!48\)\( T^{2} + \)\(56\!\cdots\!18\)\( T^{4} - \)\(37\!\cdots\!48\)\( p^{24} T^{6} + p^{48} T^{8} \)
89$D_{4}$ \( ( 1 - 319335230268 T + \)\(42\!\cdots\!98\)\( T^{2} - 319335230268 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 281771021500 T + \)\(26\!\cdots\!98\)\( T^{2} + 281771021500 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.38862583764681184418586327641, −6.97158392125510294733695556463, −6.69212059926974678475296611508, −6.47635433076558344361903560219, −6.36284231789522751146916501417, −5.69229090410818306906285054685, −5.68963986316932647434832276507, −5.46667831992611307368472112736, −5.29678492980335223069778434722, −4.82221007143909600331536553246, −4.46652654023053635256043675757, −4.42603446198817933599147759786, −3.85414690677547123035609625151, −3.47784393643271737147984651967, −3.31732693013190666805325696762, −3.12768227998746587934670783764, −2.46907053700895336089944503897, −2.15964989152490578065586746236, −2.11233084009995956619397243784, −2.07163672527353475261551872104, −1.51408953848780337305056239318, −0.991016630432321507126857616823, −0.828513463807411141679349920857, −0.48741523832051420899938464015, −0.45719984181686033732246664342, 0.45719984181686033732246664342, 0.48741523832051420899938464015, 0.828513463807411141679349920857, 0.991016630432321507126857616823, 1.51408953848780337305056239318, 2.07163672527353475261551872104, 2.11233084009995956619397243784, 2.15964989152490578065586746236, 2.46907053700895336089944503897, 3.12768227998746587934670783764, 3.31732693013190666805325696762, 3.47784393643271737147984651967, 3.85414690677547123035609625151, 4.42603446198817933599147759786, 4.46652654023053635256043675757, 4.82221007143909600331536553246, 5.29678492980335223069778434722, 5.46667831992611307368472112736, 5.68963986316932647434832276507, 5.69229090410818306906285054685, 6.36284231789522751146916501417, 6.47635433076558344361903560219, 6.69212059926974678475296611508, 6.97158392125510294733695556463, 7.38862583764681184418586327641

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