Properties

Label 8-5070e4-1.1-c1e4-0-2
Degree $8$
Conductor $6.607\times 10^{14}$
Sign $1$
Analytic cond. $2.68621\times 10^{6}$
Root an. cond. $6.36271$
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  − 4·3-s − 2·4-s + 10·9-s + 8·12-s + 3·16-s + 16·17-s − 2·25-s − 20·27-s − 8·29-s − 20·36-s + 24·43-s − 12·48-s + 10·49-s − 64·51-s − 8·53-s − 16·61-s − 4·64-s − 32·68-s + 8·75-s + 40·79-s + 35·81-s + 32·87-s + 4·100-s + 8·101-s + 16·103-s − 24·107-s + 40·108-s + ⋯
L(s)  = 1  − 2.30·3-s − 4-s + 10/3·9-s + 2.30·12-s + 3/4·16-s + 3.88·17-s − 2/5·25-s − 3.84·27-s − 1.48·29-s − 3.33·36-s + 3.65·43-s − 1.73·48-s + 10/7·49-s − 8.96·51-s − 1.09·53-s − 2.04·61-s − 1/2·64-s − 3.88·68-s + 0.923·75-s + 4.50·79-s + 35/9·81-s + 3.43·87-s + 2/5·100-s + 0.796·101-s + 1.57·103-s − 2.32·107-s + 3.84·108-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(2.68621\times 10^{6}\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.299451173\)
\(L(\frac12)\) \(\approx\) \(2.299451173\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_1$ \( ( 1 + T )^{4} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
13 \( 1 \)
good7$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 30 T^{2} + 419 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
19$D_4\times C_2$ \( 1 - 2 p T^{2} + 891 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 20 T^{2} + 1254 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 50 T^{2} + 3171 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
47$D_4\times C_2$ \( 1 - 146 T^{2} + 9315 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 4 T + 107 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 84 T^{2} + 5654 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 236 T^{2} + 22710 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 116 T^{2} + 6534 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 236 T^{2} + 25974 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 54 T^{2} + 14219 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 164 T^{2} + 23814 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
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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

−5.76272516776063713651467379375, −5.43605345475495981303458471354, −5.41706139362196146515190430013, −5.40757975314726165289853241647, −5.28072999593660479153679792147, −4.82625416314967211969721486630, −4.80080800642675224849020508396, −4.42372900458056467662082465513, −4.19198226656784861686529904621, −4.18439127943731529974593344757, −3.97506642576963185699042054390, −3.66361966625167060608170690836, −3.49164937411602068940364658509, −3.34844700392920738627505690887, −3.05661034651306125954233263108, −2.76049507478416843875073529866, −2.62505785345482465074887341911, −2.08950087139785942106762745508, −1.78187385341771302833441685523, −1.77182527361582038994531693629, −1.19991264600636679015586514802, −1.13222105329963699464190322757, −0.812840892031614584164812983219, −0.47188832565754415054088149750, −0.46698173531477592581671371476, 0.46698173531477592581671371476, 0.47188832565754415054088149750, 0.812840892031614584164812983219, 1.13222105329963699464190322757, 1.19991264600636679015586514802, 1.77182527361582038994531693629, 1.78187385341771302833441685523, 2.08950087139785942106762745508, 2.62505785345482465074887341911, 2.76049507478416843875073529866, 3.05661034651306125954233263108, 3.34844700392920738627505690887, 3.49164937411602068940364658509, 3.66361966625167060608170690836, 3.97506642576963185699042054390, 4.18439127943731529974593344757, 4.19198226656784861686529904621, 4.42372900458056467662082465513, 4.80080800642675224849020508396, 4.82625416314967211969721486630, 5.28072999593660479153679792147, 5.40757975314726165289853241647, 5.41706139362196146515190430013, 5.43605345475495981303458471354, 5.76272516776063713651467379375

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