Properties

Label 8-5070e4-1.1-c1e4-0-3
Degree $8$
Conductor $6.607\times 10^{14}$
Sign $1$
Analytic cond. $2.68621\times 10^{6}$
Root an. cond. $6.36271$
Motivic weight $1$
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  − 4·2-s + 4·3-s + 10·4-s + 4·5-s − 16·6-s − 2·7-s − 20·8-s + 10·9-s − 16·10-s + 2·11-s + 40·12-s + 8·14-s + 16·15-s + 35·16-s + 16·17-s − 40·18-s + 40·20-s − 8·21-s − 8·22-s + 4·23-s − 80·24-s + 10·25-s + 20·27-s − 20·28-s + 8·29-s − 64·30-s − 4·31-s + ⋯
L(s)  = 1  − 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s − 0.755·7-s − 7.07·8-s + 10/3·9-s − 5.05·10-s + 0.603·11-s + 11.5·12-s + 2.13·14-s + 4.13·15-s + 35/4·16-s + 3.88·17-s − 9.42·18-s + 8.94·20-s − 1.74·21-s − 1.70·22-s + 0.834·23-s − 16.3·24-s + 2·25-s + 3.84·27-s − 3.77·28-s + 1.48·29-s − 11.6·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(2.68621\times 10^{6}\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.05438598\)
\(L(\frac12)\) \(\approx\) \(14.05438598\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
3$C_1$ \( ( 1 - T )^{4} \)
5$C_1$ \( ( 1 - T )^{4} \)
13 \( 1 \)
good7$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 9 T^{2} + 22 T^{3} + 80 T^{4} + 22 p T^{5} + 9 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 10 T^{2} - 4 T^{3} + 159 T^{4} - 4 p T^{5} + 10 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 13 T^{2} + 72 T^{3} + 240 T^{4} + 72 p T^{5} + 13 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 49 T^{2} - 260 T^{3} + 1248 T^{4} - 260 p T^{5} + 49 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 77 T^{2} - 548 T^{3} + 3160 T^{4} - 548 p T^{5} + 77 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 45 T^{2} - 184 T^{3} - 88 T^{4} - 184 p T^{5} + 45 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 114 T^{2} - 680 T^{3} + 4895 T^{4} - 680 p T^{5} + 114 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 88 T^{2} + 332 T^{3} + 4686 T^{4} + 332 p T^{5} + 88 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 3 p T^{2} - 970 T^{3} + 6824 T^{4} - 970 p T^{5} + 3 p^{3} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 106 T^{2} + 352 T^{3} + 4443 T^{4} + 352 p T^{5} + 106 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 137 T^{2} - 1268 T^{3} + 8956 T^{4} - 1268 p T^{5} + 137 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 205 T^{2} - 846 T^{3} + 16944 T^{4} - 846 p T^{5} + 205 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 216 T^{2} + 2268 T^{3} + 20318 T^{4} + 2268 p T^{5} + 216 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 184 T^{2} + 1472 T^{3} + 11022 T^{4} + 1472 p T^{5} + 184 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 168 T^{2} + 900 T^{3} + 9326 T^{4} + 900 p T^{5} + 168 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 169 T^{2} + 1510 T^{3} + 17728 T^{4} + 1510 p T^{5} + 169 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 232 T^{2} + 2048 T^{3} + 19710 T^{4} + 2048 p T^{5} + 232 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 241 T^{2} - 284 T^{3} + 26052 T^{4} - 284 p T^{5} + 241 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 36 T + 768 T^{2} - 11100 T^{3} + 124766 T^{4} - 11100 p T^{5} + 768 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.99201914243998547241336205304, −5.84505304073339319997011884007, −5.61858980325906754675658191968, −5.34087396705328820126080238679, −5.17007559805906567136007275107, −4.79571733980393321871939888286, −4.52380280267014941555016734959, −4.51887431100614028282246193595, −4.12319295560417543655870025919, −3.68628622548320028319888550150, −3.52103616228626148594094929593, −3.42927013666057917774808813484, −3.29872521028188318056054497034, −2.83985346275550562577635992449, −2.76131231924620139097082609279, −2.73051132904940938365469557864, −2.71817608386031528643137313268, −1.96546230718536684133851008700, −1.93504206259727481857262766316, −1.71454879044491847523544570217, −1.66212441934029509137155483068, −1.05653749710363503837799324999, −1.04655167751832239974043186229, −0.70895536765787174477528688077, −0.68609541638473039354441200815, 0.68609541638473039354441200815, 0.70895536765787174477528688077, 1.04655167751832239974043186229, 1.05653749710363503837799324999, 1.66212441934029509137155483068, 1.71454879044491847523544570217, 1.93504206259727481857262766316, 1.96546230718536684133851008700, 2.71817608386031528643137313268, 2.73051132904940938365469557864, 2.76131231924620139097082609279, 2.83985346275550562577635992449, 3.29872521028188318056054497034, 3.42927013666057917774808813484, 3.52103616228626148594094929593, 3.68628622548320028319888550150, 4.12319295560417543655870025919, 4.51887431100614028282246193595, 4.52380280267014941555016734959, 4.79571733980393321871939888286, 5.17007559805906567136007275107, 5.34087396705328820126080238679, 5.61858980325906754675658191968, 5.84505304073339319997011884007, 5.99201914243998547241336205304

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