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Properties

Label 8-90e8-1.1-c1e4-0-5
Degree $8$
Conductor $4.305\times 10^{15}$
Sign $1$
Analytic cond. $1.75004\times 10^{7}$
Root an. cond. $8.04231$
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  − 7-s + 3·11-s + 2·13-s + 9·17-s − 4·19-s − 3·23-s − 9·29-s + 2·31-s − 37-s + 9·41-s + 8·43-s + 12·47-s − 9·49-s + 12·53-s − 15·59-s − 61-s + 11·67-s + 12·71-s − 10·73-s − 3·77-s − 7·79-s + 12·83-s − 3·89-s − 2·91-s + 5·97-s + 12·101-s − 7·103-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.904·11-s + 0.554·13-s + 2.18·17-s − 0.917·19-s − 0.625·23-s − 1.67·29-s + 0.359·31-s − 0.164·37-s + 1.40·41-s + 1.21·43-s + 1.75·47-s − 9/7·49-s + 1.64·53-s − 1.95·59-s − 0.128·61-s + 1.34·67-s + 1.42·71-s − 1.17·73-s − 0.341·77-s − 0.787·79-s + 1.31·83-s − 0.317·89-s − 0.209·91-s + 0.507·97-s + 1.19·101-s − 0.689·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.75004\times 10^{7}\)
Root analytic conductor: \(8.04231\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{16} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.695100417\)
\(L(\frac12)\) \(\approx\) \(9.695100417\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2 \wr S_4$ \( 1 + T + 10 T^{2} + 31 T^{3} + 43 T^{4} + 31 p T^{5} + 10 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 3 T + 29 T^{2} - 54 T^{3} + 369 T^{4} - 54 p T^{5} + 29 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 2 T + 34 T^{2} - 77 T^{3} + 589 T^{4} - 77 p T^{5} + 34 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 9 T + 80 T^{2} - 441 T^{3} + 2115 T^{4} - 441 p T^{5} + 80 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 4 T + 49 T^{2} + 148 T^{3} + 1117 T^{4} + 148 p T^{5} + 49 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 3 T + p T^{2} - 36 T^{3} + 27 T^{4} - 36 p T^{5} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 9 T + 56 T^{2} + 117 T^{3} + 405 T^{4} + 117 p T^{5} + 56 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 2 T + 106 T^{2} - 149 T^{3} + 4675 T^{4} - 149 p T^{5} + 106 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + T + 109 T^{2} + 88 T^{3} + 5425 T^{4} + 88 p T^{5} + 109 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 9 T + 119 T^{2} - 702 T^{3} + 6477 T^{4} - 702 p T^{5} + 119 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 8 T + 118 T^{2} - 608 T^{3} + 6487 T^{4} - 608 p T^{5} + 118 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 12 T + 137 T^{2} - 1116 T^{3} + 7461 T^{4} - 1116 p T^{5} + 137 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 12 T + 134 T^{2} - 1287 T^{3} + 11367 T^{4} - 1287 p T^{5} + 134 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 15 T + 122 T^{2} + 27 T^{3} - 2637 T^{4} + 27 p T^{5} + 122 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + T + 169 T^{2} - 128 T^{3} + 12859 T^{4} - 128 p T^{5} + 169 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 11 T + 184 T^{2} - 1613 T^{3} + 18535 T^{4} - 1613 p T^{5} + 184 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 12 T + 212 T^{2} - 1665 T^{3} + 19293 T^{4} - 1665 p T^{5} + 212 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 10 T + 280 T^{2} + 1897 T^{3} + 29707 T^{4} + 1897 p T^{5} + 280 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 7 T + p T^{2} + 10 p T^{3} + 15493 T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 12 T + 236 T^{2} - 2295 T^{3} + 29097 T^{4} - 2295 p T^{5} + 236 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 3 T + 113 T^{2} + 1314 T^{3} + 10185 T^{4} + 1314 p T^{5} + 113 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 5 T + 253 T^{2} - 788 T^{3} + 32275 T^{4} - 788 p T^{5} + 253 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.69066001353148556149577515316, −5.25129442813099548032460865136, −5.01799916430367604576739771383, −4.99788321742638863688452636472, −4.95234059939308579058750962286, −4.45517220675694073757228466947, −4.16761463009619293156030414958, −4.16170767875287173069654447859, −4.04047780471764059105027750464, −3.83483003996397274541603699765, −3.63877401264921874784845120663, −3.44154365836448778544340664962, −3.26720472363497753757162357689, −2.89761259198170092083794703100, −2.85840362987117029274852674281, −2.68497912062197213163286355143, −2.37166354133009151265083766242, −1.88299936515585409787180133940, −1.85067680552876725644737693881, −1.67625215259235229729200565122, −1.65563399792915422792173376191, −0.831177209002191274367505588560, −0.792409258528777316446321963421, −0.74850251689653631791240457775, −0.40161429012176209099891686840, 0.40161429012176209099891686840, 0.74850251689653631791240457775, 0.792409258528777316446321963421, 0.831177209002191274367505588560, 1.65563399792915422792173376191, 1.67625215259235229729200565122, 1.85067680552876725644737693881, 1.88299936515585409787180133940, 2.37166354133009151265083766242, 2.68497912062197213163286355143, 2.85840362987117029274852674281, 2.89761259198170092083794703100, 3.26720472363497753757162357689, 3.44154365836448778544340664962, 3.63877401264921874784845120663, 3.83483003996397274541603699765, 4.04047780471764059105027750464, 4.16170767875287173069654447859, 4.16761463009619293156030414958, 4.45517220675694073757228466947, 4.95234059939308579058750962286, 4.99788321742638863688452636472, 5.01799916430367604576739771383, 5.25129442813099548032460865136, 5.69066001353148556149577515316

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