Properties

Label 8-6762e4-1.1-c1e4-0-9
Degree $8$
Conductor $2.091\times 10^{15}$
Sign $1$
Analytic cond. $8.49980\times 10^{6}$
Root an. cond. $7.34811$
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 + 8·5-s + 16·6-s + 20·8-s + 10·9-s + 32·10-s + 4·11-s + 40·12-s + 12·13-s + 32·15-s + 35·16-s − 4·17-s + 40·18-s + 4·19-s + 80·20-s + 16·22-s + 4·23-s + 80·24-s + 24·25-s + 48·26-s + 20·27-s + 8·29-s + 128·30-s − 4·31-s + 56·32-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 3.57·5-s + 6.53·6-s + 7.07·8-s + 10/3·9-s + 10.1·10-s + 1.20·11-s + 11.5·12-s + 3.32·13-s + 8.26·15-s + 35/4·16-s − 0.970·17-s + 9.42·18-s + 0.917·19-s + 17.8·20-s + 3.41·22-s + 0.834·23-s + 16.3·24-s + 24/5·25-s + 9.41·26-s + 3.84·27-s + 1.48·29-s + 23.3·30-s − 0.718·31-s + 9.89·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(8.49980\times 10^{6}\)
Root analytic conductor: \(7.34811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6762} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(858.6611297\)
\(L(\frac12)\) \(\approx\) \(858.6611297\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
23$C_1$ \( ( 1 - T )^{4} \)
good5$D_{4}$ \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 24 T^{2} - 84 T^{3} + 302 T^{4} - 84 p T^{5} + 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 88 T^{2} - 452 T^{3} + 1838 T^{4} - 452 p T^{5} + 88 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 56 T^{2} + 156 T^{3} + 1310 T^{4} + 156 p T^{5} + 56 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 20 T^{2} + 52 T^{3} - 254 T^{4} + 52 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 12 T^{2} + 40 T^{3} - 202 T^{4} + 40 p T^{5} + 12 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 68 T^{2} + 412 T^{3} + 2322 T^{4} + 412 p T^{5} + 68 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 136 T^{2} - 396 T^{3} + 7310 T^{4} - 396 p T^{5} + 136 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 80 T^{2} + 8 p T^{3} + 2722 T^{4} + 8 p^{2} T^{5} + 80 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 48 T^{2} - 404 T^{3} + 3518 T^{4} - 404 p T^{5} + 48 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 132 T^{2} - 980 T^{3} + 8402 T^{4} - 980 p T^{5} + 132 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 120 T^{2} - 364 T^{3} + 8750 T^{4} - 364 p T^{5} + 120 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 204 T^{2} - 1808 T^{3} + 16086 T^{4} - 1808 p T^{5} + 204 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 232 T^{2} - 2576 T^{3} + 20866 T^{4} - 2576 p T^{5} + 232 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 304 T^{2} - 3460 T^{3} + 31774 T^{4} - 3460 p T^{5} + 304 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 444 T^{2} + 5176 T^{3} + 51542 T^{4} + 5176 p T^{5} + 444 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 224 T^{2} - 1288 T^{3} + 21730 T^{4} - 1288 p T^{5} + 224 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 32 T + 612 T^{2} + 7840 T^{3} + 1002 p T^{4} + 7840 p T^{5} + 612 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 308 T^{2} + 1004 T^{3} + 37378 T^{4} + 1004 p T^{5} + 308 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 168 T^{2} - 684 T^{3} + 46 p T^{4} - 684 p T^{5} + 168 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 184 T^{2} - 636 T^{3} + 24830 T^{4} - 636 p T^{5} + 184 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.63942980142277588928850216456, −5.42933897536444709830126028780, −5.17185446719591091642677395850, −5.11496613284156610403570399138, −5.05951355903535309563718511566, −4.43856349247759557483441101380, −4.35612191292964076911228046086, −4.25056417551811009490585248569, −4.05647369741984547701059810062, −3.73401829890525961926494350287, −3.62601175159474304772984379429, −3.62203460052559028555398306379, −3.49212564280571627776146902271, −2.99713995526637507318493328235, −2.72839653219958247413428148072, −2.67189461194578782799577199625, −2.61795309847609335892901493883, −2.27440294658378563693597071642, −2.00805285728216589094190472193, −1.99726765775455513003810426827, −1.73833058124905428566825319584, −1.23936912562065374082258074901, −1.22420228723279914382478809997, −1.22080431292357361817214059081, −0.925783858006291518621602803512, 0.925783858006291518621602803512, 1.22080431292357361817214059081, 1.22420228723279914382478809997, 1.23936912562065374082258074901, 1.73833058124905428566825319584, 1.99726765775455513003810426827, 2.00805285728216589094190472193, 2.27440294658378563693597071642, 2.61795309847609335892901493883, 2.67189461194578782799577199625, 2.72839653219958247413428148072, 2.99713995526637507318493328235, 3.49212564280571627776146902271, 3.62203460052559028555398306379, 3.62601175159474304772984379429, 3.73401829890525961926494350287, 4.05647369741984547701059810062, 4.25056417551811009490585248569, 4.35612191292964076911228046086, 4.43856349247759557483441101380, 5.05951355903535309563718511566, 5.11496613284156610403570399138, 5.17185446719591091642677395850, 5.42933897536444709830126028780, 5.63942980142277588928850216456

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