Properties

Label 8-315e4-1.1-c1e4-0-21
Degree $8$
Conductor $9845600625$
Sign $1$
Analytic cond. $40.0267$
Root an. cond. $1.58596$
Motivic weight $1$
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 + 11·4-s + 4·5-s + 22·8-s + 16·10-s − 2·11-s + 8·13-s + 36·16-s − 4·17-s + 2·19-s + 44·20-s − 8·22-s + 4·23-s + 5·25-s + 32·26-s − 12·31-s + 52·32-s − 16·34-s + 12·37-s + 8·38-s + 88·40-s − 6·43-s − 22·44-s + 16·46-s − 18·47-s − 11·49-s + 20·50-s + ⋯
L(s)  = 1  + 2.82·2-s + 11/2·4-s + 1.78·5-s + 7.77·8-s + 5.05·10-s − 0.603·11-s + 2.21·13-s + 9·16-s − 0.970·17-s + 0.458·19-s + 9.83·20-s − 1.70·22-s + 0.834·23-s + 25-s + 6.27·26-s − 2.15·31-s + 9.19·32-s − 2.74·34-s + 1.97·37-s + 1.29·38-s + 13.9·40-s − 0.914·43-s − 3.31·44-s + 2.35·46-s − 2.62·47-s − 1.57·49-s + 2.82·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(40.0267\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(19.77729414\)
\(L(\frac12)\) \(\approx\) \(19.77729414\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
good2$D_4\times C_2$ \( 1 - p^{2} T + 5 T^{2} + p T^{3} - 11 T^{4} + p^{2} T^{5} + 5 p^{2} T^{6} - p^{5} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 4 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 4 T + 20 T^{2} + 100 T^{3} + 271 T^{4} + 100 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 2 T - 32 T^{2} + 4 T^{3} + 859 T^{4} + 4 p T^{5} - 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 4 T + 53 T^{2} - 244 T^{3} + 1588 T^{4} - 244 p T^{5} + 53 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 696 p T^{5} + 106 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^3$ \( 1 - 12 T + 72 T^{2} - 288 T^{3} + 983 T^{4} - 288 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 60 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 18 T + 90 T^{2} - 528 T^{3} - 8377 T^{4} - 528 p T^{5} + 90 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
59$D_4\times C_2$ \( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 108 p T^{5} - 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 1308 p T^{5} + 157 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 22 T + 137 T^{2} - 834 T^{3} - 16648 T^{4} - 834 p T^{5} + 137 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 24 T + 144 T^{2} - 24 p T^{3} - 31057 T^{4} - 24 p^{2} T^{5} + 144 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 6 T + 148 T^{2} - 816 T^{3} + 13203 T^{4} - 816 p T^{5} + 148 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 140 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2$$\times$$C_2^2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \)
97$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} + 12 T^{3} - 8818 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.744975007304773776853400418058, −7.974913656203726401878413575969, −7.75928506069318038345276459003, −7.56259890146580419679183122821, −7.32713256083962367345237802799, −7.07562979378819235013823503953, −6.46423356562691877185709349814, −6.35771002083969034758203653722, −6.30066415976652779133915936087, −5.95451606252530015226940757024, −5.93675540895659970075214323457, −5.87407130476950780515077247836, −5.10073904849151121515528887970, −5.03112270028810569346207446293, −4.84259991501080136095445023405, −4.43473851816862788064673651876, −4.27738218916088437129536300274, −3.55170658982895102797845505484, −3.42566783995476983091508702397, −3.05564448238268555593243844399, −3.01665663749895562781597327469, −2.54158487228008431246839753345, −1.84696828558025605549915731406, −1.69014486281869312076594626809, −1.56568752475378383043535673980, 1.56568752475378383043535673980, 1.69014486281869312076594626809, 1.84696828558025605549915731406, 2.54158487228008431246839753345, 3.01665663749895562781597327469, 3.05564448238268555593243844399, 3.42566783995476983091508702397, 3.55170658982895102797845505484, 4.27738218916088437129536300274, 4.43473851816862788064673651876, 4.84259991501080136095445023405, 5.03112270028810569346207446293, 5.10073904849151121515528887970, 5.87407130476950780515077247836, 5.93675540895659970075214323457, 5.95451606252530015226940757024, 6.30066415976652779133915936087, 6.35771002083969034758203653722, 6.46423356562691877185709349814, 7.07562979378819235013823503953, 7.32713256083962367345237802799, 7.56259890146580419679183122821, 7.75928506069318038345276459003, 7.974913656203726401878413575969, 8.744975007304773776853400418058

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