Properties

Label 8-75e4-1.1-c9e4-0-2
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $2.22635\times 10^{6}$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.56e4·16-s − 3.29e7·31-s + 7.91e8·61-s − 3.87e8·81-s + 9.43e9·121-s + ⋯
L(s)  = 1  − 0.326·16-s − 6.41·31-s + 7.31·61-s − 81-s + 4·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(2.22635\times 10^{6}\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(1.271589161\)
\(L(\frac12)\) \(\approx\) \(1.271589161\)
\(L(\frac{11}{2})\) 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$C_2^2$ \( 1 + p^{18} T^{4} \)
5 \( 1 \)
good2$C_2^3$ \( 1 + 85673 T^{4} + p^{36} T^{8} \)
7$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
11$C_2$ \( ( 1 - p^{9} T^{2} )^{4} \)
13$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
17$C_2^3$ \( 1 - \)\(59\!\cdots\!42\)\( T^{4} + p^{36} T^{8} \)
19$C_2^2$ \( ( 1 + 428575456298 T^{2} + p^{18} T^{4} )^{2} \)
23$C_2^3$ \( 1 + \)\(48\!\cdots\!78\)\( T^{4} + p^{36} T^{8} \)
29$C_2$ \( ( 1 + p^{9} T^{2} )^{4} \)
31$C_2$ \( ( 1 + 8247368 T + p^{9} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
41$C_2$ \( ( 1 - p^{9} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
47$C_2^3$ \( 1 + \)\(22\!\cdots\!38\)\( T^{4} + p^{36} T^{8} \)
53$C_2^3$ \( 1 + \)\(52\!\cdots\!38\)\( T^{4} + p^{36} T^{8} \)
59$C_2$ \( ( 1 + p^{9} T^{2} )^{4} \)
61$C_2$ \( ( 1 - 197894882 T + p^{9} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
71$C_2$ \( ( 1 - p^{9} T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 61992800032369822 T^{2} + p^{18} T^{4} )^{2} \)
83$C_2^3$ \( 1 + \)\(57\!\cdots\!78\)\( T^{4} + p^{36} T^{8} \)
89$C_2$ \( ( 1 + p^{9} T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
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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

−8.813007274261545294510572573535, −8.673743464693534582471495236976, −8.416221566442216637647382441818, −7.70781653387246311363829553864, −7.51643527163353402367013206116, −7.43158229333381916323603782751, −6.90173412151997834106510944796, −6.72446625916902976959110198835, −6.54090218423603259375700624490, −5.60816764105624014732329760696, −5.60494368292015842542879028345, −5.40524694329017291118976640573, −5.25150332569875340860528679875, −4.53149901559091257778624416918, −4.08385392450591470051968928345, −3.75745047490238042825346750396, −3.67934268839615054305622414550, −3.21184237952587834073663444436, −2.68136318527357079034212681428, −2.01649066548814756855308894765, −1.90218631780234062999349876865, −1.84979377758533418799793945521, −0.907401606410959924206216905170, −0.66846494216827235557483573569, −0.17592590938500470586699811320, 0.17592590938500470586699811320, 0.66846494216827235557483573569, 0.907401606410959924206216905170, 1.84979377758533418799793945521, 1.90218631780234062999349876865, 2.01649066548814756855308894765, 2.68136318527357079034212681428, 3.21184237952587834073663444436, 3.67934268839615054305622414550, 3.75745047490238042825346750396, 4.08385392450591470051968928345, 4.53149901559091257778624416918, 5.25150332569875340860528679875, 5.40524694329017291118976640573, 5.60494368292015842542879028345, 5.60816764105624014732329760696, 6.54090218423603259375700624490, 6.72446625916902976959110198835, 6.90173412151997834106510944796, 7.43158229333381916323603782751, 7.51643527163353402367013206116, 7.70781653387246311363829553864, 8.416221566442216637647382441818, 8.673743464693534582471495236976, 8.813007274261545294510572573535

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