Properties

Label 8-55e4-1.1-c2e4-0-0
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  − 6·4-s − 8·5-s − 6·9-s − 16·11-s − 5·16-s + 48·20-s − 2·25-s − 12·31-s + 36·36-s + 96·44-s + 48·45-s + 54·49-s + 128·55-s − 72·59-s + 180·64-s + 108·71-s + 40·80-s − 135·81-s + 148·89-s + 96·99-s + 12·100-s − 50·121-s + 72·124-s + 344·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 3/2·4-s − 8/5·5-s − 2/3·9-s − 1.45·11-s − 0.312·16-s + 12/5·20-s − 0.0799·25-s − 0.387·31-s + 36-s + 2.18·44-s + 1.06·45-s + 1.10·49-s + 2.32·55-s − 1.22·59-s + 2.81·64-s + 1.52·71-s + 1/2·80-s − 5/3·81-s + 1.66·89-s + 0.969·99-s + 3/25·100-s − 0.413·121-s + 0.580·124-s + 2.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-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

\(L(\frac{3}{2})\) \(\approx\) \(0.1607258317\)
\(L(\frac12)\) \(\approx\) \(0.1607258317\)
\(L(2)\) 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$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 + 3 T^{2} + p^{4} T^{4} )^{2} \)
3$C_2^2$ \( ( 1 + p T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 27 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 258 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 333 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 617 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 302 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1577 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 3 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 2717 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2942 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 3198 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 302 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 5597 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 2297 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 8222 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 27 T + p^{2} T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + 7278 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 8702 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 8658 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 37 T + p^{2} T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 4622 T^{2} + p^{4} T^{4} )^{2} \)
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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

−11.02255395285345655254890121023, −10.91507624097873169819168341564, −10.84655233683138591611718745533, −10.21591682642774139044432991805, −9.788051607354212917151227502354, −9.689003553454867781936673285901, −9.143288824775113474728760810942, −9.025201353235607442155897659455, −8.399230419425414840925618325462, −8.345673693503228594227201495975, −8.218832410589635386792558489607, −7.54443145476151559923124113315, −7.34234694542062870537769388707, −7.22612425117650555708964636582, −6.34575058399440641653942530814, −6.14522861890324466390573263655, −5.46029289199434541157255064752, −5.20064453304599800182415505491, −4.75942501512394589558549552843, −4.44631741643833273131071602981, −3.80844486151246108552313990043, −3.74273532979832357879019884551, −2.88371272931084688676825711416, −2.26661718137789986588070035939, −0.30465384144035135398849971846, 0.30465384144035135398849971846, 2.26661718137789986588070035939, 2.88371272931084688676825711416, 3.74273532979832357879019884551, 3.80844486151246108552313990043, 4.44631741643833273131071602981, 4.75942501512394589558549552843, 5.20064453304599800182415505491, 5.46029289199434541157255064752, 6.14522861890324466390573263655, 6.34575058399440641653942530814, 7.22612425117650555708964636582, 7.34234694542062870537769388707, 7.54443145476151559923124113315, 8.218832410589635386792558489607, 8.345673693503228594227201495975, 8.399230419425414840925618325462, 9.025201353235607442155897659455, 9.143288824775113474728760810942, 9.689003553454867781936673285901, 9.788051607354212917151227502354, 10.21591682642774139044432991805, 10.84655233683138591611718745533, 10.91507624097873169819168341564, 11.02255395285345655254890121023

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