Properties

Label 8-55e4-1.1-c2e4-0-2
Degree $8$
Conductor $9150625$
Sign $1$
Analytic cond. $5.04418$
Root an. cond. $1.22419$
Motivic weight $2$
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  + 5·2-s − 6·3-s + 16·4-s + 5·5-s − 30·6-s + 30·7-s + 30·8-s + 29·9-s + 25·10-s − 11-s − 96·12-s − 30·13-s + 150·14-s − 30·15-s + 20·16-s − 50·17-s + 145·18-s − 45·19-s + 80·20-s − 180·21-s − 5·22-s − 16·23-s − 180·24-s + 10·25-s − 150·26-s − 80·27-s + 480·28-s + ⋯
L(s)  = 1  + 5/2·2-s − 2·3-s + 4·4-s + 5-s − 5·6-s + 30/7·7-s + 15/4·8-s + 29/9·9-s + 5/2·10-s − 0.0909·11-s − 8·12-s − 2.30·13-s + 75/7·14-s − 2·15-s + 5/4·16-s − 2.94·17-s + 8.05·18-s − 2.36·19-s + 4·20-s − 8.57·21-s − 0.227·22-s − 0.695·23-s − 7.5·24-s + 2/5·25-s − 5.76·26-s − 2.96·27-s + 17.1·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(9150625\)    =    \(5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5.04418\)
Root analytic conductor: \(1.22419\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 9150625,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.916814593\)
\(L(\frac12)\) \(\approx\) \(4.916814593\)
\(L(2)\) 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$C_4$ \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
11$C_4$ \( 1 + T - 19 p T^{2} + p^{2} T^{3} + p^{4} T^{4} \)
good2$C_2^2:C_4$ \( 1 - 5 T + 9 T^{2} + 5 T^{3} - 39 T^{4} + 5 p^{2} T^{5} + 9 p^{4} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} \)
3$C_2^2:C_4$ \( 1 + 2 p T + 7 T^{2} - 52 T^{3} - 275 T^{4} - 52 p^{2} T^{5} + 7 p^{4} T^{6} + 2 p^{7} T^{7} + p^{8} T^{8} \)
7$C_2^2:C_4$ \( 1 - 30 T + 67 p T^{2} - 720 p T^{3} + 40501 T^{4} - 720 p^{3} T^{5} + 67 p^{5} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \)
13$C_2^2:C_4$ \( 1 + 30 T + 529 T^{2} + 7230 T^{3} + 95881 T^{4} + 7230 p^{2} T^{5} + 529 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \)
17$C_2^2:C_4$ \( 1 + 50 T + 67 p T^{2} + 900 p T^{3} + 204721 T^{4} + 900 p^{3} T^{5} + 67 p^{5} T^{6} + 50 p^{6} T^{7} + p^{8} T^{8} \)
19$C_2^2:C_4$ \( 1 + 45 T + 721 T^{2} - 6165 T^{3} - 348164 T^{4} - 6165 p^{2} T^{5} + 721 p^{4} T^{6} + 45 p^{6} T^{7} + p^{8} T^{8} \)
23$D_{4}$ \( ( 1 + 8 T + 1054 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 + 70 T + 2801 T^{2} + 86310 T^{3} + 2547721 T^{4} + 86310 p^{2} T^{5} + 2801 p^{4} T^{6} + 70 p^{6} T^{7} + p^{8} T^{8} \)
31$C_2^2:C_4$ \( 1 - 98 T + 3963 T^{2} - 112216 T^{3} + 3333125 T^{4} - 112216 p^{2} T^{5} + 3963 p^{4} T^{6} - 98 p^{6} T^{7} + p^{8} T^{8} \)
37$C_2^2:C_4$ \( 1 - 16 T + 687 T^{2} - 1274 p T^{3} + 2737805 T^{4} - 1274 p^{3} T^{5} + 687 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \)
41$C_2^2:C_4$ \( 1 - 40 T + 2681 T^{2} - 149760 T^{3} + 5656561 T^{4} - 149760 p^{2} T^{5} + 2681 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \)
43$C_2^2:C_4$ \( 1 - 3071 T^{2} + 9079081 T^{4} - 3071 p^{4} T^{6} + p^{8} T^{8} \)
47$C_2^2:C_4$ \( 1 + 24 T - 33 T^{2} - 75608 T^{3} - 440895 T^{4} - 75608 p^{2} T^{5} - 33 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \)
53$C_2^2:C_4$ \( 1 + 2 p T + 1827 T^{2} - 298792 T^{3} - 25913995 T^{4} - 298792 p^{2} T^{5} + 1827 p^{4} T^{6} + 2 p^{7} T^{7} + p^{8} T^{8} \)
59$C_2^2:C_4$ \( 1 - 217 T + 19353 T^{2} - 973679 T^{3} + 45202700 T^{4} - 973679 p^{2} T^{5} + 19353 p^{4} T^{6} - 217 p^{6} T^{7} + p^{8} T^{8} \)
61$C_2^2:C_4$ \( 1 + 80 T + 3241 T^{2} - 355430 T^{3} - 32613299 T^{4} - 355430 p^{2} T^{5} + 3241 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 - 33 T + 8469 T^{2} - 33 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2:C_4$ \( 1 - 38 T - 2677 T^{2} + 351464 T^{3} + 725125 T^{4} + 351464 p^{2} T^{5} - 2677 p^{4} T^{6} - 38 p^{6} T^{7} + p^{8} T^{8} \)
73$C_2^2:C_4$ \( 1 + 230 T + 29479 T^{2} + 2908180 T^{3} + 232776841 T^{4} + 2908180 p^{2} T^{5} + 29479 p^{4} T^{6} + 230 p^{6} T^{7} + p^{8} T^{8} \)
79$C_2^2:C_4$ \( 1 + 100 T + 2641 T^{2} - 506200 T^{3} - 68278319 T^{4} - 506200 p^{2} T^{5} + 2641 p^{4} T^{6} + 100 p^{6} T^{7} + p^{8} T^{8} \)
83$C_2^2:C_4$ \( 1 - 185 T + 25759 T^{2} - 3065635 T^{3} + 270642136 T^{4} - 3065635 p^{2} T^{5} + 25759 p^{4} T^{6} - 185 p^{6} T^{7} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 - 201 T + 25881 T^{2} - 201 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$C_2^2:C_4$ \( 1 - 101 T - 1683 T^{2} - 513883 T^{3} + 138713780 T^{4} - 513883 p^{2} T^{5} - 1683 p^{4} T^{6} - 101 p^{6} T^{7} + p^{8} T^{8} \)
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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

−11.17022490984381353349027799473, −11.15781388648187002978562957018, −11.04521197788173260052866296289, −10.51975607648803704563832735546, −10.41613845871478874354855331480, −9.877970409563187072439655448471, −9.443373707798003607495143971736, −8.839353634070666182908988373964, −8.603378971194490212670846904344, −8.006225393439554798219308319896, −7.83705542010846166137037372040, −7.27343686543474534188545626503, −6.77304477131095890429685486451, −6.75806087826619127982953720635, −6.15683293127585680532299158307, −6.00974600726600110805714136669, −5.59733902058064771381814299854, −4.77396529515831548768909366872, −4.75535116237023738824848257483, −4.68551260800110522275531849983, −4.65918404822299105240183814582, −4.08207161700978121689888146120, −2.21868371067105211748862598734, −2.19643230649410577860036406552, −2.03349421732793223270346297428, 2.03349421732793223270346297428, 2.19643230649410577860036406552, 2.21868371067105211748862598734, 4.08207161700978121689888146120, 4.65918404822299105240183814582, 4.68551260800110522275531849983, 4.75535116237023738824848257483, 4.77396529515831548768909366872, 5.59733902058064771381814299854, 6.00974600726600110805714136669, 6.15683293127585680532299158307, 6.75806087826619127982953720635, 6.77304477131095890429685486451, 7.27343686543474534188545626503, 7.83705542010846166137037372040, 8.006225393439554798219308319896, 8.603378971194490212670846904344, 8.839353634070666182908988373964, 9.443373707798003607495143971736, 9.877970409563187072439655448471, 10.41613845871478874354855331480, 10.51975607648803704563832735546, 11.04521197788173260052866296289, 11.15781388648187002978562957018, 11.17022490984381353349027799473

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