Properties

Label 8-1323e4-1.1-c1e4-0-4
Degree $8$
Conductor $3.064\times 10^{12}$
Sign $1$
Analytic cond. $12455.1$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 8·5-s − 20·17-s + 16·20-s + 30·25-s + 8·37-s − 8·41-s − 12·47-s − 20·59-s − 36·67-s + 40·68-s − 28·79-s − 8·83-s + 160·85-s − 20·89-s − 60·100-s − 28·101-s + 48·109-s − 30·121-s − 80·125-s + 127-s + 131-s + 137-s + 139-s − 16·148-s + 149-s + 151-s + ⋯
L(s)  = 1  − 4-s − 3.57·5-s − 4.85·17-s + 3.57·20-s + 6·25-s + 1.31·37-s − 1.24·41-s − 1.75·47-s − 2.60·59-s − 4.39·67-s + 4.85·68-s − 3.15·79-s − 0.878·83-s + 17.3·85-s − 2.11·89-s − 6·100-s − 2.78·101-s + 4.59·109-s − 2.72·121-s − 7.15·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.31·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{12} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(12455.1\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1323} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{12} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good2$C_4\times C_2$ \( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} \)
5$D_{4}$ \( ( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 + 30 T^{2} + 422 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 + 46 T^{2} + 862 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
19$D_4\times C_2$ \( 1 + 30 T^{2} + 902 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 56 T^{2} + 1522 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 10 T^{2} + 1582 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 + 70 T^{2} + 2902 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 4 T + p T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 81 T^{2} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 96 T^{2} + p^{2} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 230 T^{2} + 20622 T^{4} + 230 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 18 T + 210 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + 68 T^{2} + 7318 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 + 112 T^{2} + 5794 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 14 T + 187 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 + 4 T + 165 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 358 T^{2} + 50734 T^{4} + 358 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.37676777238209853721802696721, −7.27030697103457993748613057820, −6.85815223054740220300002676493, −6.68241344916331689584017786868, −6.53530563103237893414746977830, −6.47045923616832105901722655340, −5.95606302773223515319506685285, −5.80862355191575090664748702731, −5.67620303428959603353526329794, −4.87331069501633146061756233145, −4.81786313234275237617708244101, −4.76754311643787982761360149825, −4.63333547884182629727588088166, −4.22620981602061599362381339749, −4.16039531196809668671815621033, −4.11805687308438026586540756681, −3.94943155852185593928110096552, −3.37552809113310081065867172898, −3.23955540029834902077943282500, −2.86706676752173035794115106752, −2.78486206050463560543656389938, −2.37399114869500803124800628435, −1.90124082908638249499907997659, −1.52706886126793220429518408191, −1.19008522522013261609864204473, 0, 0, 0, 0, 1.19008522522013261609864204473, 1.52706886126793220429518408191, 1.90124082908638249499907997659, 2.37399114869500803124800628435, 2.78486206050463560543656389938, 2.86706676752173035794115106752, 3.23955540029834902077943282500, 3.37552809113310081065867172898, 3.94943155852185593928110096552, 4.11805687308438026586540756681, 4.16039531196809668671815621033, 4.22620981602061599362381339749, 4.63333547884182629727588088166, 4.76754311643787982761360149825, 4.81786313234275237617708244101, 4.87331069501633146061756233145, 5.67620303428959603353526329794, 5.80862355191575090664748702731, 5.95606302773223515319506685285, 6.47045923616832105901722655340, 6.53530563103237893414746977830, 6.68241344916331689584017786868, 6.85815223054740220300002676493, 7.27030697103457993748613057820, 7.37676777238209853721802696721

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