Properties

Label 8-6762e4-1.1-c1e4-0-10
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 $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 4·3-s + 10·4-s − 3·5-s − 16·6-s + 20·8-s + 10·9-s − 12·10-s + 2·11-s − 40·12-s − 5·13-s + 12·15-s + 35·16-s − 10·17-s + 40·18-s − 8·19-s − 30·20-s + 8·22-s − 4·23-s − 80·24-s + 25-s − 20·26-s − 20·27-s + 3·29-s + 48·30-s + 6·31-s + 56·32-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 5·4-s − 1.34·5-s − 6.53·6-s + 7.07·8-s + 10/3·9-s − 3.79·10-s + 0.603·11-s − 11.5·12-s − 1.38·13-s + 3.09·15-s + 35/4·16-s − 2.42·17-s + 9.42·18-s − 1.83·19-s − 6.70·20-s + 1.70·22-s − 0.834·23-s − 16.3·24-s + 1/5·25-s − 3.92·26-s − 3.84·27-s + 0.557·29-s + 8.76·30-s + 1.07·31-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: \(4\)
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

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{4} \)
3$C_1$ \( ( 1 + T )^{4} \)
7 \( 1 \)
23$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 3 T + 8 T^{2} + 13 T^{3} + 14 T^{4} + 13 p T^{5} + 8 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 2 T + 20 T^{2} - 6 p T^{3} + 230 T^{4} - 6 p^{2} T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 5 T + 46 T^{2} + 175 T^{3} + 850 T^{4} + 175 p T^{5} + 46 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 10 T + 64 T^{2} + 326 T^{3} + 1534 T^{4} + 326 p T^{5} + 64 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
29$C_2 \wr S_4$ \( 1 - 3 T + 10 T^{2} - 105 T^{3} + 1642 T^{4} - 105 p T^{5} + 10 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 6 T + 32 T^{2} - 262 T^{3} + 2526 T^{4} - 262 p T^{5} + 32 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - T + 2 p T^{2} + 205 T^{3} + 2410 T^{4} + 205 p T^{5} + 2 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + T + 84 T^{2} - 5 T^{3} + 4550 T^{4} - 5 p T^{5} + 84 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + T + 44 T^{2} + 9 T^{3} + 3558 T^{4} + 9 p T^{5} + 44 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 17 T + 186 T^{2} + 1449 T^{3} + 10610 T^{4} + 1449 p T^{5} + 186 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 4 T + 124 T^{2} - 92 T^{3} + 6870 T^{4} - 92 p T^{5} + 124 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 12 T^{2} + 128 T^{3} + 5462 T^{4} + 128 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 24 T + 404 T^{2} + 4600 T^{3} + 41910 T^{4} + 4600 p T^{5} + 404 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 8 T + 164 T^{2} + 440 T^{3} + 10182 T^{4} + 440 p T^{5} + 164 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 4 T + 124 T^{2} + 212 T^{3} + 11110 T^{4} + 212 p T^{5} + 124 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 6 T + 156 T^{2} + 834 T^{3} + 11990 T^{4} + 834 p T^{5} + 156 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 6 T + 180 T^{2} - 942 T^{3} + 15830 T^{4} - 942 p T^{5} + 180 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 18 T + 320 T^{2} + 3226 T^{3} + 34958 T^{4} + 3226 p T^{5} + 320 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 26 T + 568 T^{2} + 7462 T^{3} + 85294 T^{4} + 7462 p T^{5} + 568 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 13 T + 42 T^{2} - 1165 T^{3} - 14182 T^{4} - 1165 p T^{5} + 42 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.07082996003083589615265270577, −5.59474440635654953690077425946, −5.50874753543384454169406578765, −5.49917491340100099267445805055, −5.31729564532094992504864233480, −4.81552737757104401961765908020, −4.63918801775060075912512557588, −4.55622416418944421976212204810, −4.53449067684653035382903269130, −4.51672205191371788564253426017, −4.25559087899004267513693077360, −4.13531101339035048897633406660, −3.96149356923393788835711542590, −3.51569374475604985635268536291, −3.47297623075049083184693589518, −3.27287710794807709521989652806, −2.94842273704218926092446544845, −2.66399943960803245393479694401, −2.31497319533247266143115321025, −2.29755703023574502075135501736, −2.28567286846969769122911738192, −1.65963618854498590650889768048, −1.39341535780073484769676425262, −1.37831191702156250099468176372, −1.08795194773697262277374519416, 0, 0, 0, 0, 1.08795194773697262277374519416, 1.37831191702156250099468176372, 1.39341535780073484769676425262, 1.65963618854498590650889768048, 2.28567286846969769122911738192, 2.29755703023574502075135501736, 2.31497319533247266143115321025, 2.66399943960803245393479694401, 2.94842273704218926092446544845, 3.27287710794807709521989652806, 3.47297623075049083184693589518, 3.51569374475604985635268536291, 3.96149356923393788835711542590, 4.13531101339035048897633406660, 4.25559087899004267513693077360, 4.51672205191371788564253426017, 4.53449067684653035382903269130, 4.55622416418944421976212204810, 4.63918801775060075912512557588, 4.81552737757104401961765908020, 5.31729564532094992504864233480, 5.49917491340100099267445805055, 5.50874753543384454169406578765, 5.59474440635654953690077425946, 6.07082996003083589615265270577

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