Properties

Label 8-768e4-1.1-c1e4-0-15
Degree $8$
Conductor $347892350976$
Sign $1$
Analytic cond. $1414.33$
Root an. cond. $2.47639$
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  + 4·13-s + 24·17-s + 24·29-s + 20·37-s + 24·49-s − 24·53-s + 20·61-s − 81-s − 48·97-s − 24·101-s − 52·109-s + 24·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 1.10·13-s + 5.82·17-s + 4.45·29-s + 3.28·37-s + 24/7·49-s − 3.29·53-s + 2.56·61-s − 1/9·81-s − 4.87·97-s − 2.38·101-s − 4.98·109-s + 2.25·113-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 + 8/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1414.33\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{768} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.105631884\)
\(L(\frac12)\) \(\approx\) \(6.105631884\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2^2$ \( 1 + T^{4} \)
good5$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
19$C_2^3$ \( 1 - 718 T^{4} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 - 1198 T^{4} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 4946 T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 156 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 13294 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
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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

−7.64340962057668306588341134608, −7.04643869292729180050602723186, −6.96908556305762567160143693356, −6.83736371930147637848370698229, −6.50120315332585576670052625241, −6.00059774001320690001200293524, −5.88179680798172814230354886252, −5.85361778133456208325362938869, −5.82260377270653208016572957990, −5.27204372899415693772699156815, −4.98230126632478189209828918158, −4.89721548869852060331115232352, −4.69565340696397159593973456623, −3.99438193009910738850352172580, −3.91661395583101381765092522252, −3.82199307781898194440601318929, −3.52810934061880463504169612801, −2.97706935708064055533661501402, −2.69245372217153760571364097139, −2.66733736921756499512389558293, −2.65824867182734389364088723837, −1.30932783198403752609688096531, −1.25118953822207777146799686642, −1.17120054947929185734665838804, −0.853137126994992542397206426901, 0.853137126994992542397206426901, 1.17120054947929185734665838804, 1.25118953822207777146799686642, 1.30932783198403752609688096531, 2.65824867182734389364088723837, 2.66733736921756499512389558293, 2.69245372217153760571364097139, 2.97706935708064055533661501402, 3.52810934061880463504169612801, 3.82199307781898194440601318929, 3.91661395583101381765092522252, 3.99438193009910738850352172580, 4.69565340696397159593973456623, 4.89721548869852060331115232352, 4.98230126632478189209828918158, 5.27204372899415693772699156815, 5.82260377270653208016572957990, 5.85361778133456208325362938869, 5.88179680798172814230354886252, 6.00059774001320690001200293524, 6.50120315332585576670052625241, 6.83736371930147637848370698229, 6.96908556305762567160143693356, 7.04643869292729180050602723186, 7.64340962057668306588341134608

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