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Properties

Label 8-75e4-1.1-c2e4-0-4
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $17.4415$
Root an. cond. $1.42954$
Motivic weight $2$
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  + 4·2-s + 8·4-s − 4·7-s + 12·8-s + 16·11-s + 32·13-s − 16·14-s + 15·16-s + 40·17-s + 64·22-s − 56·23-s + 128·26-s − 32·28-s − 16·31-s + 40·32-s + 160·34-s − 64·37-s − 56·41-s + 8·43-s + 128·44-s − 224·46-s − 128·47-s + 8·49-s + 256·52-s − 56·53-s − 48·56-s + 200·61-s + ⋯
L(s)  = 1  + 2·2-s + 2·4-s − 4/7·7-s + 3/2·8-s + 1.45·11-s + 2.46·13-s − 8/7·14-s + 0.937·16-s + 2.35·17-s + 2.90·22-s − 2.43·23-s + 4.92·26-s − 8/7·28-s − 0.516·31-s + 5/4·32-s + 4.70·34-s − 1.72·37-s − 1.36·41-s + 8/43·43-s + 2.90·44-s − 4.86·46-s − 2.72·47-s + 8/49·49-s + 4.92·52-s − 1.05·53-s − 6/7·56-s + 3.27·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(17.4415\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(6.106015812\)
\(L(\frac12)\) \(\approx\) \(6.106015812\)
\(L(2)\) 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$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
good2$D_4\times C_2$ \( 1 - p^{2} T + p^{3} T^{2} - 3 p^{2} T^{3} + 17 T^{4} - 3 p^{4} T^{5} + p^{7} T^{6} - p^{8} T^{7} + p^{8} T^{8} \)
7$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} + 156 T^{3} + 2942 T^{4} + 156 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \)
11$D_{4}$ \( ( 1 - 8 T + 204 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 32 T + 512 T^{2} - 9120 T^{3} + 148994 T^{4} - 9120 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 40 T + 800 T^{2} - 15240 T^{3} + 281858 T^{4} - 15240 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 940 T^{2} + 450438 T^{4} - 940 p^{4} T^{6} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 + 56 T + 1568 T^{2} + 50904 T^{3} + 1508162 T^{4} + 50904 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 - 2128 T^{2} + 2165634 T^{4} - 2128 p^{4} T^{6} + p^{8} T^{8} \)
31$D_{4}$ \( ( 1 + 8 T + 1722 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 64 T + 2048 T^{2} + 58176 T^{3} + 1440962 T^{4} + 58176 p^{2} T^{5} + 2048 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} \)
41$D_{4}$ \( ( 1 + 28 T + 3342 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 5256 T^{3} - 557566 T^{4} - 5256 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 + 128 T + 8192 T^{2} + 506496 T^{3} + 28260194 T^{4} + 506496 p^{2} T^{5} + 8192 p^{4} T^{6} + 128 p^{6} T^{7} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 + 56 T + 1568 T^{2} + 155064 T^{3} + 15333122 T^{4} + 155064 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 + 200 T^{2} - 5646222 T^{4} + 200 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 - 100 T + 7998 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 200 T + 20000 T^{2} - 1888200 T^{3} + 153742658 T^{4} - 1888200 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} \)
71$C_2$ \( ( 1 + 68 T + p^{2} T^{2} )^{4} \)
73$D_4\times C_2$ \( 1 + 76 T + 2888 T^{2} - 65436 T^{3} - 36833458 T^{4} - 65436 p^{2} T^{5} + 2888 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8} \)
79$C_2^2$ \( ( 1 - 11882 T^{2} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 16 T + 128 T^{2} - 101328 T^{3} + 79904642 T^{4} - 101328 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 16060 T^{2} + 188845638 T^{4} - 16060 p^{4} T^{6} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 20 T + 200 T^{2} - 173820 T^{3} + 150551438 T^{4} - 173820 p^{2} T^{5} + 200 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \)
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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

−10.52795810329223048353131285238, −10.26902422828552830293315898281, −10.05251592367988167095496601382, −9.669996670677761813852262975587, −9.660903649745329383884856412377, −8.927527622430037280535679498836, −8.566505160652055425661068781340, −8.402810876226042059373920286862, −8.162838991593512893350707408004, −7.70333963418304722356852670782, −7.25391599946350552797624846492, −6.85357534504635271669409153445, −6.41027452808946318855785491258, −6.31740806299076602911839864052, −5.92699959048462790542498919912, −5.75241657961393025445355604351, −5.07430020921464277112897775116, −5.06853230785096259032629481286, −4.27828341536192349585621990593, −3.69029309320002129007462159245, −3.67929523041263810703640580517, −3.61282662090980943200106223891, −2.92671815863290521531326806479, −1.77407759404490910354620685529, −1.36898646714398648337005601066, 1.36898646714398648337005601066, 1.77407759404490910354620685529, 2.92671815863290521531326806479, 3.61282662090980943200106223891, 3.67929523041263810703640580517, 3.69029309320002129007462159245, 4.27828341536192349585621990593, 5.06853230785096259032629481286, 5.07430020921464277112897775116, 5.75241657961393025445355604351, 5.92699959048462790542498919912, 6.31740806299076602911839864052, 6.41027452808946318855785491258, 6.85357534504635271669409153445, 7.25391599946350552797624846492, 7.70333963418304722356852670782, 8.162838991593512893350707408004, 8.402810876226042059373920286862, 8.566505160652055425661068781340, 8.927527622430037280535679498836, 9.660903649745329383884856412377, 9.669996670677761813852262975587, 10.05251592367988167095496601382, 10.26902422828552830293315898281, 10.52795810329223048353131285238

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