Properties

Label 8-6762e4-1.1-c1e4-0-6
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 + 6·5-s − 16·6-s + 20·8-s + 10·9-s + 24·10-s − 40·12-s + 10·13-s − 24·15-s + 35·16-s + 12·17-s + 40·18-s + 8·19-s + 60·20-s − 4·23-s − 80·24-s + 11·25-s + 40·26-s − 20·27-s + 6·29-s − 96·30-s + 8·31-s + 56·32-s + 48·34-s + 100·36-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 5·4-s + 2.68·5-s − 6.53·6-s + 7.07·8-s + 10/3·9-s + 7.58·10-s − 11.5·12-s + 2.77·13-s − 6.19·15-s + 35/4·16-s + 2.91·17-s + 9.42·18-s + 1.83·19-s + 13.4·20-s − 0.834·23-s − 16.3·24-s + 11/5·25-s + 7.84·26-s − 3.84·27-s + 1.11·29-s − 17.5·30-s + 1.43·31-s + 9.89·32-s + 8.23·34-s + 50/3·36-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

\(L(1)\) \(\approx\) \(125.0575316\)
\(L(\frac12)\) \(\approx\) \(125.0575316\)
\(L(\frac{3}{2})\) 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$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + p^{2} T^{2} - 78 T^{3} + 202 T^{4} - 78 p T^{5} + p^{4} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 26 T^{2} + 8 T^{3} + 386 T^{4} + 8 p T^{5} + 26 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 71 T^{2} - 2 p^{2} T^{3} + 1384 T^{4} - 2 p^{3} T^{5} + 71 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 88 T^{2} - 516 T^{3} + 2446 T^{4} - 516 p T^{5} + 88 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 103 T^{2} - 390 T^{3} + 4096 T^{4} - 390 p T^{5} + 103 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 - 4 T + 64 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 127 T^{2} + 622 T^{3} + 6656 T^{4} + 622 p T^{5} + 127 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 195 T^{2} - 1734 T^{3} + 12626 T^{4} - 1734 p T^{5} + 195 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 125 T^{2} + 606 T^{3} + 7708 T^{4} + 606 p T^{5} + 125 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 165 T^{2} - 242 T^{3} + 11090 T^{4} - 242 p T^{5} + 165 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 134 T^{2} + 164 T^{3} + 8002 T^{4} + 164 p T^{5} + 134 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 218 T^{2} - 1240 T^{3} + 18538 T^{4} - 1240 p T^{5} + 218 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 212 T^{2} - 2140 T^{3} + 18546 T^{4} - 2140 p T^{5} + 212 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 224 T^{2} + 772 T^{3} + 21102 T^{4} + 772 p T^{5} + 224 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 266 T^{2} - 8 T^{3} + 27746 T^{4} - 8 p T^{5} + 266 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 276 T^{2} - 2316 T^{3} + 402 p T^{4} - 2316 p T^{5} + 276 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 514 T^{2} + 6328 T^{3} + 69386 T^{4} + 6328 p T^{5} + 514 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 330 T^{2} - 3528 T^{3} + 40994 T^{4} - 3528 p T^{5} + 330 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 262 T^{2} - 2076 T^{3} + 31042 T^{4} - 2076 p T^{5} + 262 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 30 T + 601 T^{2} - 8294 T^{3} + 91892 T^{4} - 8294 p T^{5} + 601 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.68440953513841601105471791722, −5.36292828176366729023406637159, −5.31590468473931080106215787480, −5.24686856499377897241529361914, −5.14175490160518899970121119110, −4.69069084795955345197958000281, −4.52984011965150928870617580493, −4.50418696561405992020877470848, −4.17603727561138088557855957574, −3.80539204266639407336499035892, −3.69639913686244611501712053690, −3.61688126153777712584343268895, −3.54437812802650912824602900762, −3.03078201733257928878567711719, −3.01934364909721724632491477467, −2.69446748416167835478368692646, −2.60402264845893627692030673159, −2.03365831166524500584554889129, −1.85111588495953664357878254988, −1.78021533957523535667322885787, −1.64304882668991452988966266633, −1.12209246768043369894249605601, −0.975477757198303226772163186207, −0.903814998778094796770615592642, −0.72359501173465975687925293271, 0.72359501173465975687925293271, 0.903814998778094796770615592642, 0.975477757198303226772163186207, 1.12209246768043369894249605601, 1.64304882668991452988966266633, 1.78021533957523535667322885787, 1.85111588495953664357878254988, 2.03365831166524500584554889129, 2.60402264845893627692030673159, 2.69446748416167835478368692646, 3.01934364909721724632491477467, 3.03078201733257928878567711719, 3.54437812802650912824602900762, 3.61688126153777712584343268895, 3.69639913686244611501712053690, 3.80539204266639407336499035892, 4.17603727561138088557855957574, 4.50418696561405992020877470848, 4.52984011965150928870617580493, 4.69069084795955345197958000281, 5.14175490160518899970121119110, 5.24686856499377897241529361914, 5.31590468473931080106215787480, 5.36292828176366729023406637159, 5.68440953513841601105471791722

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