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Properties

Label 8-4025e4-1.1-c1e4-0-1
Degree $8$
Conductor $2.625\times 10^{14}$
Sign $1$
Analytic cond. $1.06701\times 10^{6}$
Root an. cond. $5.66919$
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  − 2-s − 4·3-s − 2·4-s + 4·6-s − 4·7-s + 3·8-s + 5·9-s + 11·11-s + 8·12-s + 3·13-s + 4·14-s − 2·16-s − 13·17-s − 5·18-s + 8·19-s + 16·21-s − 11·22-s + 4·23-s − 12·24-s − 3·26-s + 3·27-s + 8·28-s − 2·29-s + 8·31-s − 44·33-s + 13·34-s − 10·36-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s − 4-s + 1.63·6-s − 1.51·7-s + 1.06·8-s + 5/3·9-s + 3.31·11-s + 2.30·12-s + 0.832·13-s + 1.06·14-s − 1/2·16-s − 3.15·17-s − 1.17·18-s + 1.83·19-s + 3.49·21-s − 2.34·22-s + 0.834·23-s − 2.44·24-s − 0.588·26-s + 0.577·27-s + 1.51·28-s − 0.371·29-s + 1.43·31-s − 7.65·33-s + 2.22·34-s − 5/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1.06701\times 10^{6}\)
Root analytic conductor: \(5.66919\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.103864334\)
\(L(\frac12)\) \(\approx\) \(1.103864334\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
23$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + T + 3 T^{2} + p T^{3} + 7 T^{4} + p^{2} T^{5} + 3 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
3$C_2 \wr S_4$ \( 1 + 4 T + 11 T^{2} + 7 p T^{3} + 40 T^{4} + 7 p^{2} T^{5} + 11 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - p T + 84 T^{2} - 38 p T^{3} + 1630 T^{4} - 38 p^{2} T^{5} + 84 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 3 T + 42 T^{2} - 94 T^{3} + 752 T^{4} - 94 p T^{5} + 42 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 13 T + 126 T^{2} + 764 T^{3} + 3760 T^{4} + 764 p T^{5} + 126 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 8 T + 79 T^{2} - 375 T^{3} + 2172 T^{4} - 375 p T^{5} + 79 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 2 T + 64 T^{2} + 302 T^{3} + 1982 T^{4} + 302 p T^{5} + 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 8 T + 127 T^{2} - 721 T^{3} + 5960 T^{4} - 721 p T^{5} + 127 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 5 T + 76 T^{2} - 504 T^{3} + 3412 T^{4} - 504 p T^{5} + 76 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 23 T + 277 T^{2} - 2246 T^{3} + 15230 T^{4} - 2246 p T^{5} + 277 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 2 T + 55 T^{2} + 103 T^{3} + 1724 T^{4} + 103 p T^{5} + 55 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 2 T + 117 T^{2} + 73 T^{3} + 6904 T^{4} + 73 p T^{5} + 117 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - T + 150 T^{2} - 104 T^{3} + 10648 T^{4} - 104 p T^{5} + 150 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 15 T + 286 T^{2} - 2620 T^{3} + 26558 T^{4} - 2620 p T^{5} + 286 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - T + 76 T^{2} - 168 T^{3} + 6712 T^{4} - 168 p T^{5} + 76 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 5 T + 264 T^{2} - 984 T^{3} + 26394 T^{4} - 984 p T^{5} + 264 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 8 T - 15 T^{2} - 323 T^{3} + 964 T^{4} - 323 p T^{5} - 15 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 3 T + 108 T^{2} + 542 T^{3} + 12212 T^{4} + 542 p T^{5} + 108 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 169 T^{2} - 121 T^{3} + 17656 T^{4} - 121 p T^{5} + 169 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 5 T + 162 T^{2} + 1438 T^{3} + 15118 T^{4} + 1438 p T^{5} + 162 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 35 T + 810 T^{2} - 11932 T^{3} + 133804 T^{4} - 11932 p T^{5} + 810 p^{2} T^{6} - 35 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 11 T + 177 T^{2} + 318 T^{3} + 2406 T^{4} + 318 p T^{5} + 177 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.11266244866416570296086402609, −5.82318842267777224254394498410, −5.53105457531215180404643024277, −5.45253750994907193587455611087, −5.41492091510618828392727950231, −4.81771432404343022892202497992, −4.69822590083957559185164638284, −4.59603325109918561091445690085, −4.53145418945493691317453134929, −4.18242404733025471155728464779, −3.93777127073194765418349758569, −3.93355209171796828657464362395, −3.71210250277671881389471034876, −3.29380920925126133305539775642, −3.22367488961690249823672395549, −2.76799492928821368871467827515, −2.59200840372728108829927398784, −2.37531558036257145057000501787, −1.96815415344754670443461596461, −1.64572695335736017348186143471, −1.32931135407640514794443682738, −0.896475112333114795476241847727, −0.70183762148836059839169105810, −0.58427531140249799268196442897, −0.45276816533256622590101738162, 0.45276816533256622590101738162, 0.58427531140249799268196442897, 0.70183762148836059839169105810, 0.896475112333114795476241847727, 1.32931135407640514794443682738, 1.64572695335736017348186143471, 1.96815415344754670443461596461, 2.37531558036257145057000501787, 2.59200840372728108829927398784, 2.76799492928821368871467827515, 3.22367488961690249823672395549, 3.29380920925126133305539775642, 3.71210250277671881389471034876, 3.93355209171796828657464362395, 3.93777127073194765418349758569, 4.18242404733025471155728464779, 4.53145418945493691317453134929, 4.59603325109918561091445690085, 4.69822590083957559185164638284, 4.81771432404343022892202497992, 5.41492091510618828392727950231, 5.45253750994907193587455611087, 5.53105457531215180404643024277, 5.82318842267777224254394498410, 6.11266244866416570296086402609

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