Properties

Label 8-75e4-1.1-c17e4-0-1
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $3.56579\times 10^{8}$
Root an. cond. $11.7224$
Motivic weight $17$
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  − 8.92e4·4-s − 8.60e7·9-s − 2.26e9·11-s + 6.47e9·16-s + 3.88e11·19-s + 9.47e12·29-s + 4.51e12·31-s + 7.68e12·36-s − 2.61e13·41-s + 2.01e14·44-s + 5.96e14·49-s − 1.92e15·59-s − 5.71e15·61-s − 1.97e15·64-s + 4.41e15·71-s − 3.46e16·76-s − 9.99e15·79-s + 5.55e15·81-s + 1.21e17·89-s + 1.94e17·99-s − 3.10e16·101-s + 6.02e17·109-s − 8.45e17·116-s + 1.53e18·121-s − 4.02e17·124-s + ⋯
L(s)  = 1  − 0.680·4-s − 2/3·9-s − 3.18·11-s + 0.376·16-s + 5.25·19-s + 3.51·29-s + 0.950·31-s + 0.453·36-s − 0.510·41-s + 2.16·44-s + 2.56·49-s − 1.70·59-s − 3.81·61-s − 0.878·64-s + 0.811·71-s − 3.57·76-s − 0.741·79-s + 1/3·81-s + 3.27·89-s + 2.12·99-s − 0.285·101-s + 2.89·109-s − 2.39·116-s + 3.03·121-s − 0.646·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(3.56579\times 10^{8}\)
Root analytic conductor: \(11.7224\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :17/2, 17/2, 17/2, 17/2),\ 1)\)

Particular Values

\(L(9)\) \(\approx\) \(3.684526283\)
\(L(\frac12)\) \(\approx\) \(3.684526283\)
\(L(\frac{19}{2})\) 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)$
bad3$C_2$ \( ( 1 + p^{16} T^{2} )^{2} \)
5 \( 1 \)
good2$D_4\times C_2$ \( 1 + 5577 p^{4} T^{2} + 363353 p^{12} T^{4} + 5577 p^{38} T^{6} + p^{68} T^{8} \)
7$D_4\times C_2$ \( 1 - 12173272053916 p^{2} T^{2} + \)\(71\!\cdots\!78\)\( p^{4} T^{4} - 12173272053916 p^{36} T^{6} + p^{68} T^{8} \)
11$D_{4}$ \( ( 1 + 1131629912 T + 1154204164608501734 T^{2} + 1131629912 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 16040368211818558412 T^{2} + \)\(12\!\cdots\!42\)\( p^{2} T^{4} - 16040368211818558412 p^{34} T^{6} + p^{68} T^{8} \)
17$D_4\times C_2$ \( 1 - \)\(30\!\cdots\!60\)\( T^{2} + \)\(37\!\cdots\!58\)\( T^{4} - \)\(30\!\cdots\!60\)\( p^{34} T^{6} + p^{68} T^{8} \)
19$D_{4}$ \( ( 1 - 194376127216 T + \)\(19\!\cdots\!58\)\( T^{2} - 194376127216 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - \)\(52\!\cdots\!36\)\( T^{2} + \)\(10\!\cdots\!98\)\( T^{4} - \)\(52\!\cdots\!36\)\( p^{34} T^{6} + p^{68} T^{8} \)
29$D_{4}$ \( ( 1 - 4737062625892 T + \)\(13\!\cdots\!18\)\( T^{2} - 4737062625892 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 2256407074872 T + \)\(39\!\cdots\!22\)\( T^{2} - 2256407074872 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - \)\(45\!\cdots\!24\)\( T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - \)\(45\!\cdots\!24\)\( p^{34} T^{6} + p^{68} T^{8} \)
41$D_{4}$ \( ( 1 + 13051545859228 T + \)\(52\!\cdots\!42\)\( T^{2} + 13051545859228 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(19\!\cdots\!80\)\( T^{2} + \)\(16\!\cdots\!98\)\( T^{4} - \)\(19\!\cdots\!80\)\( p^{34} T^{6} + p^{68} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(82\!\cdots\!20\)\( T^{2} + \)\(30\!\cdots\!38\)\( T^{4} - \)\(82\!\cdots\!20\)\( p^{34} T^{6} + p^{68} T^{8} \)
53$D_4\times C_2$ \( 1 + \)\(23\!\cdots\!84\)\( T^{2} + \)\(85\!\cdots\!78\)\( T^{4} + \)\(23\!\cdots\!84\)\( p^{34} T^{6} + p^{68} T^{8} \)
59$D_{4}$ \( ( 1 + 963392423116456 T + \)\(25\!\cdots\!18\)\( T^{2} + 963392423116456 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 2858793214972660 T + \)\(65\!\cdots\!18\)\( T^{2} + 2858793214972660 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(20\!\cdots\!60\)\( T^{2} + \)\(24\!\cdots\!58\)\( T^{4} - \)\(20\!\cdots\!60\)\( p^{34} T^{6} + p^{68} T^{8} \)
71$D_{4}$ \( ( 1 - 2206536655060304 T - \)\(67\!\cdots\!14\)\( T^{2} - 2206536655060304 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(44\!\cdots\!96\)\( T^{2} + \)\(45\!\cdots\!02\)\( p^{2} T^{4} - \)\(44\!\cdots\!96\)\( p^{34} T^{6} + p^{68} T^{8} \)
79$D_{4}$ \( ( 1 + 4997891097934440 T + \)\(27\!\cdots\!18\)\( T^{2} + 4997891097934440 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(55\!\cdots\!52\)\( T^{2} + \)\(19\!\cdots\!38\)\( T^{4} - \)\(55\!\cdots\!52\)\( p^{34} T^{6} + p^{68} T^{8} \)
89$D_{4}$ \( ( 1 - 60857257652216796 T + \)\(33\!\cdots\!78\)\( T^{2} - 60857257652216796 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(99\!\cdots\!80\)\( T^{2} + \)\(51\!\cdots\!38\)\( T^{4} - \)\(99\!\cdots\!80\)\( p^{34} T^{6} + p^{68} T^{8} \)
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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

−7.56210041349466735311224029013, −7.41319365882501706429226012637, −7.20578275912888039736064264619, −6.43583289089818584552884870565, −6.27200065620536567375818759154, −5.98241752213198458527040316097, −5.61006862042028971602528961143, −5.21303942947783564523447738849, −5.17344317349882825359931123842, −5.01046039966343453728592714115, −4.61126877706638396075016004139, −4.57326410071302493317017254564, −3.85176058422371472561984552908, −3.40727672662514515557199272915, −3.07629128501760485231213282952, −3.07103670889386720000210344691, −2.78615926810744916287178118543, −2.61168557499666695458067421241, −2.28028544094859737286009888415, −1.60305692580446779807407068797, −1.19119118920844315047201689968, −1.14195786633012488828626678627, −0.69873428775041787930569473586, −0.54186718734903349866426691956, −0.24157148347029332841601968967, 0.24157148347029332841601968967, 0.54186718734903349866426691956, 0.69873428775041787930569473586, 1.14195786633012488828626678627, 1.19119118920844315047201689968, 1.60305692580446779807407068797, 2.28028544094859737286009888415, 2.61168557499666695458067421241, 2.78615926810744916287178118543, 3.07103670889386720000210344691, 3.07629128501760485231213282952, 3.40727672662514515557199272915, 3.85176058422371472561984552908, 4.57326410071302493317017254564, 4.61126877706638396075016004139, 5.01046039966343453728592714115, 5.17344317349882825359931123842, 5.21303942947783564523447738849, 5.61006862042028971602528961143, 5.98241752213198458527040316097, 6.27200065620536567375818759154, 6.43583289089818584552884870565, 7.20578275912888039736064264619, 7.41319365882501706429226012637, 7.56210041349466735311224029013

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