Properties

Label 8-1014e4-1.1-c1e4-0-10
Degree $8$
Conductor $1.057\times 10^{12}$
Sign $1$
Analytic cond. $4297.93$
Root an. cond. $2.84549$
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 + 4·23-s + 6·25-s + 20·27-s − 4·29-s − 20·36-s + 4·43-s + 12·48-s + 20·49-s + 64·51-s − 12·53-s − 16·61-s − 4·64-s − 32·68-s + 16·69-s + 24·75-s − 24·79-s + 35·81-s − 16·87-s − 8·92-s − 12·100-s + 20·101-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 + 0.834·23-s + 6/5·25-s + 3.84·27-s − 0.742·29-s − 3.33·36-s + 0.609·43-s + 1.73·48-s + 20/7·49-s + 8.96·51-s − 1.64·53-s − 2.04·61-s − 1/2·64-s − 3.88·68-s + 1.92·69-s + 2.77·75-s − 2.70·79-s + 35/9·81-s − 1.71·87-s − 0.834·92-s − 6/5·100-s + 1.99·101-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(4297.93\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1014} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.59146251\)
\(L(\frac12)\) \(\approx\) \(10.59146251\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
13 \( 1 \)
good5$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
7$D_4\times C_2$ \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 2 T + 47 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 26 T^{2} + 15 p T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 + 54 T^{2} + 3659 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 8 T + 111 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 28 T^{2} + 8586 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 260 T^{2} + 26874 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 228 T^{2} + 24074 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.41600766659694072758545971290, −7.12903314737018077618991785470, −6.77446366200483207911655818405, −6.66023388774610233276104910896, −5.95079464788934486621511728807, −5.94543613475437193748447700463, −5.82474506822383070728023347452, −5.66949197814614667171169777090, −5.21746276857318611007526588655, −4.83122794061019655331900479219, −4.77553777286908720560315081016, −4.61927499362037629232911607305, −4.30515398541692729113142390795, −3.88682029868656428418782459149, −3.68097718961713866078474745502, −3.40517161812166722841184429088, −3.23496183705299522066278176048, −3.17494057756455682483117360790, −2.85851884625716434580661568016, −2.50179221108861253382008985472, −2.10440536492097668068243114687, −1.72325044848115878292867966154, −1.34206699551082744999465520542, −0.974084463606856009848919241523, −0.75742029301580306283461037136, 0.75742029301580306283461037136, 0.974084463606856009848919241523, 1.34206699551082744999465520542, 1.72325044848115878292867966154, 2.10440536492097668068243114687, 2.50179221108861253382008985472, 2.85851884625716434580661568016, 3.17494057756455682483117360790, 3.23496183705299522066278176048, 3.40517161812166722841184429088, 3.68097718961713866078474745502, 3.88682029868656428418782459149, 4.30515398541692729113142390795, 4.61927499362037629232911607305, 4.77553777286908720560315081016, 4.83122794061019655331900479219, 5.21746276857318611007526588655, 5.66949197814614667171169777090, 5.82474506822383070728023347452, 5.94543613475437193748447700463, 5.95079464788934486621511728807, 6.66023388774610233276104910896, 6.77446366200483207911655818405, 7.12903314737018077618991785470, 7.41600766659694072758545971290

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