Properties

Label 8-700e4-1.1-c1e4-0-0
Degree $8$
Conductor $240100000000$
Sign $1$
Analytic cond. $976.114$
Root an. cond. $2.36421$
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  − 5·9-s − 12·11-s + 16·19-s − 12·29-s − 4·31-s − 12·41-s − 11·49-s + 2·61-s − 8·79-s + 9·81-s − 6·89-s + 60·99-s + 6·101-s + 34·109-s + 58·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s − 80·171-s + 173-s + ⋯
L(s)  = 1  − 5/3·9-s − 3.61·11-s + 3.67·19-s − 2.22·29-s − 0.718·31-s − 1.87·41-s − 1.57·49-s + 0.256·61-s − 0.900·79-s + 81-s − 0.635·89-s + 6.03·99-s + 0.597·101-s + 3.25·109-s + 5.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s − 6.11·171-s + 0.0760·173-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(976.114\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{700} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1577051060\)
\(L(\frac12)\) \(\approx\) \(0.1577051060\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
good3$C_2^3$ \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \)
23$C_2^3$ \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^3$ \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^3$ \( 1 - 38 T^{2} - 1365 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
67$C_2^3$ \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 - 157 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.59029294036418735482971328542, −7.49935966474912430085120076445, −7.06370523361702915577230407559, −6.98605285245511176583640423195, −6.61210937356520402815524906658, −6.32452547653195436148859367660, −5.86278475541282963043007932529, −5.65052428073291418042029720139, −5.51708881811588923150174182914, −5.41237621230457878230579942949, −5.30768511409901177601284251375, −4.98274831252056889652300491816, −4.85594022073743344940773153822, −4.43450097858714459320649525872, −4.02787335606104075141367167746, −3.44523461624446216444515653620, −3.33116525699630870877050130879, −3.22838470323681510562797052722, −3.00430670907121122626317071299, −2.57691839692257883958147439744, −2.43051744339291963771087778339, −1.82369861939299468177614442616, −1.67106804890505790399825074717, −0.838381786191964814041901911863, −0.12857275858498555271702710163, 0.12857275858498555271702710163, 0.838381786191964814041901911863, 1.67106804890505790399825074717, 1.82369861939299468177614442616, 2.43051744339291963771087778339, 2.57691839692257883958147439744, 3.00430670907121122626317071299, 3.22838470323681510562797052722, 3.33116525699630870877050130879, 3.44523461624446216444515653620, 4.02787335606104075141367167746, 4.43450097858714459320649525872, 4.85594022073743344940773153822, 4.98274831252056889652300491816, 5.30768511409901177601284251375, 5.41237621230457878230579942949, 5.51708881811588923150174182914, 5.65052428073291418042029720139, 5.86278475541282963043007932529, 6.32452547653195436148859367660, 6.61210937356520402815524906658, 6.98605285245511176583640423195, 7.06370523361702915577230407559, 7.49935966474912430085120076445, 7.59029294036418735482971328542

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