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Properties

Label 8-90e8-1.1-c1e4-0-6
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

\(L(1)\) \(\approx\) \(7.494518345\)
\(L(\frac12)\) \(\approx\) \(7.494518345\)
\(L(\frac{3}{2})\) 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

−5.41171755477601194090580281258, −5.37004449885489574547356138611, −5.11305426821942581369212421595, −4.94400866793047750618557508824, −4.94324656103658913933673177475, −4.35982124137082331071038669384, −4.34288752100145765608839884103, −4.33010210961787817089678907288, −4.18298916026199224734370487273, −3.70251227430602146089908453671, −3.58598132058437278761070238627, −3.28950757934832563436475636448, −3.28414530152519071355487632969, −2.92932260951505390905241743135, −2.84424498915974634648161498579, −2.52397711704541811017392930317, −2.50281924970209891448838097383, −1.98263775884753316316679081937, −1.87389194093565742748624513489, −1.65874169577518538026712957340, −1.61751184025072961088304789210, −0.968716165381294732876522228342, −0.805846800473531848631276903511, −0.50657145423755016829801082622, −0.43587311795446973631098246856, 0.43587311795446973631098246856, 0.50657145423755016829801082622, 0.805846800473531848631276903511, 0.968716165381294732876522228342, 1.61751184025072961088304789210, 1.65874169577518538026712957340, 1.87389194093565742748624513489, 1.98263775884753316316679081937, 2.50281924970209891448838097383, 2.52397711704541811017392930317, 2.84424498915974634648161498579, 2.92932260951505390905241743135, 3.28414530152519071355487632969, 3.28950757934832563436475636448, 3.58598132058437278761070238627, 3.70251227430602146089908453671, 4.18298916026199224734370487273, 4.33010210961787817089678907288, 4.34288752100145765608839884103, 4.35982124137082331071038669384, 4.94324656103658913933673177475, 4.94400866793047750618557508824, 5.11305426821942581369212421595, 5.37004449885489574547356138611, 5.41171755477601194090580281258

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