Properties

Label 8-525e4-1.1-c1e4-0-6
Degree $8$
Conductor $75969140625$
Sign $1$
Analytic cond. $308.848$
Root an. cond. $2.04747$
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  − 8·4-s + 6·9-s + 40·16-s − 48·36-s − 11·49-s − 160·64-s + 16·79-s + 27·81-s − 76·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 240·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 88·196-s + ⋯
L(s)  = 1  − 4·4-s + 2·9-s + 10·16-s − 8·36-s − 1.57·49-s − 20·64-s + 1.80·79-s + 3·81-s − 7.27·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 20·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 44/7·196-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(308.848\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5139687587\)
\(L(\frac12)\) \(\approx\) \(0.5139687587\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
5 \( 1 \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
good2$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
83$C_2$ \( ( 1 - p T^{2} )^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 169 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.892373657994132158383364893483, −7.72472803048816042952177769385, −7.52691468897359460854277318247, −7.31909911075668038044554405328, −6.94528768279055183720331696645, −6.48180853456505477535957617944, −6.27331866249875387612168106742, −6.26650531315286869363913951717, −5.65537697477364257038411022252, −5.34934078652998875859280284817, −5.25319537156480419360196590242, −5.05521707132237474578327744846, −4.69970713442226506322571479400, −4.65708942210780292188596659207, −4.23950650292507199390242448515, −3.97788750817692412371426635244, −3.95588915906764318521666195230, −3.63816324703329082798230012468, −3.30527053522407803150577570583, −2.98904242763748093802211547049, −2.40628662692044119453696030807, −1.65346617475158190382470573438, −1.32405775916972774205726664311, −1.02685279558057878079671248462, −0.33090966372630186139671030315, 0.33090966372630186139671030315, 1.02685279558057878079671248462, 1.32405775916972774205726664311, 1.65346617475158190382470573438, 2.40628662692044119453696030807, 2.98904242763748093802211547049, 3.30527053522407803150577570583, 3.63816324703329082798230012468, 3.95588915906764318521666195230, 3.97788750817692412371426635244, 4.23950650292507199390242448515, 4.65708942210780292188596659207, 4.69970713442226506322571479400, 5.05521707132237474578327744846, 5.25319537156480419360196590242, 5.34934078652998875859280284817, 5.65537697477364257038411022252, 6.26650531315286869363913951717, 6.27331866249875387612168106742, 6.48180853456505477535957617944, 6.94528768279055183720331696645, 7.31909911075668038044554405328, 7.52691468897359460854277318247, 7.72472803048816042952177769385, 7.892373657994132158383364893483

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