Properties

Label 8-2790e4-1.1-c1e4-0-2
Degree $8$
Conductor $6.059\times 10^{13}$
Sign $1$
Analytic cond. $246334.$
Root an. cond. $4.71998$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 10·4-s + 4·5-s + 5·7-s + 20·8-s + 16·10-s + 3·11-s + 6·13-s + 20·14-s + 35·16-s + 7·19-s + 40·20-s + 12·22-s + 23-s + 10·25-s + 24·26-s + 50·28-s + 4·29-s + 4·31-s + 56·32-s + 20·35-s + 6·37-s + 28·38-s + 80·40-s + 4·41-s + 13·43-s + 30·44-s + ⋯
L(s)  = 1  + 2.82·2-s + 5·4-s + 1.78·5-s + 1.88·7-s + 7.07·8-s + 5.05·10-s + 0.904·11-s + 1.66·13-s + 5.34·14-s + 35/4·16-s + 1.60·19-s + 8.94·20-s + 2.55·22-s + 0.208·23-s + 2·25-s + 4.70·26-s + 9.44·28-s + 0.742·29-s + 0.718·31-s + 9.89·32-s + 3.38·35-s + 0.986·37-s + 4.54·38-s + 12.6·40-s + 0.624·41-s + 1.98·43-s + 4.52·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(246334.\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2790} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(166.1706738\)
\(L(\frac12)\) \(\approx\) \(166.1706738\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
31$C_1$ \( ( 1 - T )^{4} \)
good7$C_2 \wr S_4$ \( 1 - 5 T + 24 T^{2} - 81 T^{3} + 254 T^{4} - 81 p T^{5} + 24 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 3 T + 34 T^{2} - 79 T^{3} + 530 T^{4} - 79 p T^{5} + 34 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 6 T + 40 T^{2} - 178 T^{3} + 766 T^{4} - 178 p T^{5} + 40 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 28 T^{2} - 80 T^{3} + 422 T^{4} - 80 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 7 T + 64 T^{2} - 351 T^{3} + 1774 T^{4} - 351 p T^{5} + 64 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - T + 56 T^{2} - 149 T^{3} + 1470 T^{4} - 149 p T^{5} + 56 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 4 T + 20 T^{2} - 92 T^{3} + 1014 T^{4} - 92 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$S_4\times C_2$ \( 1 - 6 T + 48 T^{2} - 290 T^{3} + 3086 T^{4} - 290 p T^{5} + 48 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 4 T + 68 T^{2} - 236 T^{3} + 3750 T^{4} - 236 p T^{5} + 68 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 13 T + 104 T^{2} - 925 T^{3} + 7870 T^{4} - 925 p T^{5} + 104 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 4 T + 76 T^{2} + 244 T^{3} + 4262 T^{4} + 244 p T^{5} + 76 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 5 T + 174 T^{2} + 727 T^{3} + 12802 T^{4} + 727 p T^{5} + 174 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 8 T + 84 T^{2} - 568 T^{3} + 6022 T^{4} - 568 p T^{5} + 84 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 4 T - 12 T^{2} + 348 T^{3} + 6470 T^{4} + 348 p T^{5} - 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 8 T + 108 T^{2} + 360 T^{3} - 58 T^{4} + 360 p T^{5} + 108 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + T + 186 T^{2} + 185 T^{3} + 18466 T^{4} + 185 p T^{5} + 186 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 15 T + 256 T^{2} - 2149 T^{3} + 23710 T^{4} - 2149 p T^{5} + 256 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 5 T + 160 T^{2} - 1393 T^{3} + 12734 T^{4} - 1393 p T^{5} + 160 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 2 T + 188 T^{2} + 1138 T^{3} + 16662 T^{4} + 1138 p T^{5} + 188 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 9 T + 118 T^{2} - 329 T^{3} + 1394 T^{4} - 329 p T^{5} + 118 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 10 T + 304 T^{2} - 2070 T^{3} + 40030 T^{4} - 2070 p T^{5} + 304 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.24847270844437533579931408716, −5.80709522272591331901787750516, −5.75592209411163678872768190974, −5.62208549487750440643783066328, −5.52757077701372068293301620199, −5.08403396394650857313952908025, −4.88817682095446517320274453253, −4.85162038705062987917400113106, −4.83580789590666717890633900747, −4.41394022223565056661905164620, −4.06164754535383577559097947569, −4.06012088800858460777235723180, −3.95265611165973164667766803785, −3.36327695980103789039237874797, −3.23874562900593166797347529024, −3.16326301803830065754091981754, −3.00214744359262292638309022445, −2.32770932960329609134253904636, −2.22380631197448290793020880110, −2.22114055419574764647340607535, −2.01552195172813676360518813606, −1.28671956258211986939822070237, −1.21060833244101024550659073358, −1.11169820000781077634929584710, −1.04274942947633929106119482628, 1.04274942947633929106119482628, 1.11169820000781077634929584710, 1.21060833244101024550659073358, 1.28671956258211986939822070237, 2.01552195172813676360518813606, 2.22114055419574764647340607535, 2.22380631197448290793020880110, 2.32770932960329609134253904636, 3.00214744359262292638309022445, 3.16326301803830065754091981754, 3.23874562900593166797347529024, 3.36327695980103789039237874797, 3.95265611165973164667766803785, 4.06012088800858460777235723180, 4.06164754535383577559097947569, 4.41394022223565056661905164620, 4.83580789590666717890633900747, 4.85162038705062987917400113106, 4.88817682095446517320274453253, 5.08403396394650857313952908025, 5.52757077701372068293301620199, 5.62208549487750440643783066328, 5.75592209411163678872768190974, 5.80709522272591331901787750516, 6.24847270844437533579931408716

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