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Properties

Label 8-2960e4-1.1-c0e4-0-2
Degree $8$
Conductor $7.677\times 10^{13}$
Sign $1$
Analytic cond. $4.76206$
Root an. cond. $1.21541$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 10·25-s + 4·37-s + 4·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 16·185-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 4·5-s + 10·25-s + 4·37-s + 4·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 16·185-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(4.76206\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5745538405\)
\(L(\frac12)\) \(\approx\) \(0.5745538405\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{4} \)
37$C_1$ \( ( 1 - T )^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
7$C_4\times C_2$ \( 1 + T^{8} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_4\times C_2$ \( 1 + T^{8} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4\times C_2$ \( 1 + T^{8} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
59$C_4\times C_2$ \( 1 + T^{8} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_4\times C_2$ \( 1 + T^{8} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_4\times C_2$ \( 1 + T^{8} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−6.42998392522518549549698932918, −6.10776852951792242339453684111, −6.08901120087902065114952433881, −5.87615811484858734475111059774, −5.62427050168799586563501094122, −5.22552537334318415479525002581, −4.92156194188483444095968565643, −4.87255282185914887143149322183, −4.65956509553633570492550124523, −4.48436201765019024269960547547, −4.30405196587916787546234091616, −4.14193346383975944259031030981, −3.86021285792515153135185902792, −3.77714881974312610658071579691, −3.34621182423389083777388520277, −3.33737701116877980251195144040, −3.19988319498128694092824081980, −2.65681745254138831216974915311, −2.60076582922549783578647103272, −2.48149730921107612621481186658, −1.98309025619997055537227703382, −1.33010157346009607046084894090, −1.20348283280735079119879897543, −0.73720671527621487263616743786, −0.50549037690951651395956898947, 0.50549037690951651395956898947, 0.73720671527621487263616743786, 1.20348283280735079119879897543, 1.33010157346009607046084894090, 1.98309025619997055537227703382, 2.48149730921107612621481186658, 2.60076582922549783578647103272, 2.65681745254138831216974915311, 3.19988319498128694092824081980, 3.33737701116877980251195144040, 3.34621182423389083777388520277, 3.77714881974312610658071579691, 3.86021285792515153135185902792, 4.14193346383975944259031030981, 4.30405196587916787546234091616, 4.48436201765019024269960547547, 4.65956509553633570492550124523, 4.87255282185914887143149322183, 4.92156194188483444095968565643, 5.22552537334318415479525002581, 5.62427050168799586563501094122, 5.87615811484858734475111059774, 6.08901120087902065114952433881, 6.10776852951792242339453684111, 6.42998392522518549549698932918

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