Properties

Label 8-490e4-1.1-c1e4-0-2
Degree $8$
Conductor $57648010000$
Sign $1$
Analytic cond. $234.364$
Root an. cond. $1.97804$
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-s − 4·5-s − 6·9-s − 6·11-s + 10·19-s − 4·20-s + 5·25-s + 16·29-s − 4·31-s − 6·36-s − 12·41-s − 6·44-s + 24·45-s + 24·55-s + 8·59-s + 12·61-s − 64-s − 24·71-s + 10·76-s + 28·79-s + 9·81-s − 4·89-s − 40·95-s + 36·99-s + 5·100-s + 4·109-s + 16·116-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.78·5-s − 2·9-s − 1.80·11-s + 2.29·19-s − 0.894·20-s + 25-s + 2.97·29-s − 0.718·31-s − 36-s − 1.87·41-s − 0.904·44-s + 3.57·45-s + 3.23·55-s + 1.04·59-s + 1.53·61-s − 1/8·64-s − 2.84·71-s + 1.14·76-s + 3.15·79-s + 81-s − 0.423·89-s − 4.10·95-s + 3.61·99-s + 1/2·100-s + 0.383·109-s + 1.48·116-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(234.364\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{490} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6860283541\)
\(L(\frac12)\) \(\approx\) \(0.6860283541\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good3$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
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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

−8.041820099794990786829978957298, −7.78411371435771016112859780838, −7.38014902304940038563148187634, −7.35075183883875532309252744906, −6.91120559008235542510847857684, −6.85409811869460402915917281336, −6.67131875261339353571096160873, −6.05405266467559695997352493370, −5.80867994051678738572591072072, −5.64696656141744043016714341504, −5.62989690338705083596775032238, −5.03938790806980925456731354160, −4.89241747775404754213773638305, −4.62703886472197383041144228885, −4.60000049404850850128388095999, −3.84995930337431560083307506826, −3.49116095007911587172906467177, −3.35069069715510860054288926039, −3.30458746709726134917505246674, −2.66847739324534965920257532387, −2.61609608103738851102870849671, −2.35521198192347087251404604157, −1.58691477413178640108174232887, −0.896651004109062042349604060252, −0.34832747559439487807399429975, 0.34832747559439487807399429975, 0.896651004109062042349604060252, 1.58691477413178640108174232887, 2.35521198192347087251404604157, 2.61609608103738851102870849671, 2.66847739324534965920257532387, 3.30458746709726134917505246674, 3.35069069715510860054288926039, 3.49116095007911587172906467177, 3.84995930337431560083307506826, 4.60000049404850850128388095999, 4.62703886472197383041144228885, 4.89241747775404754213773638305, 5.03938790806980925456731354160, 5.62989690338705083596775032238, 5.64696656141744043016714341504, 5.80867994051678738572591072072, 6.05405266467559695997352493370, 6.67131875261339353571096160873, 6.85409811869460402915917281336, 6.91120559008235542510847857684, 7.35075183883875532309252744906, 7.38014902304940038563148187634, 7.78411371435771016112859780838, 8.041820099794990786829978957298

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