Properties

Label 8-90e8-1.1-c1e4-0-2
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  + 4·19-s − 24·29-s − 4·31-s + 24·41-s + 20·49-s − 24·59-s − 16·61-s + 24·71-s + 16·79-s + 24·101-s + 4·109-s − 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 0.917·19-s − 4.45·29-s − 0.718·31-s + 3.74·41-s + 20/7·49-s − 3.12·59-s − 2.04·61-s + 2.84·71-s + 1.80·79-s + 2.38·101-s + 0.383·109-s − 3.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-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: induced by $\chi_{8100} (1, \cdot )$
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

\(L(1)\) \(\approx\) \(1.457607946\)
\(L(\frac12)\) \(\approx\) \(1.457607946\)
\(L(\frac{3}{2})\) 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$D_4\times C_2$ \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 44 T^{2} + 954 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 92 T^{2} + 4746 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 116 T^{2} + 6762 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 44 T^{2} + 5130 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
67$D_4\times C_2$ \( 1 - 20 T^{2} + 2166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 188 T^{2} + 16794 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 20 T^{2} + 8586 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 380 T^{2} + 54906 T^{4} - 380 p^{2} T^{6} + p^{4} T^{8} \)
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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

−5.57102585267548162113678575558, −5.43373247788282536580030267928, −5.14229331087035426504742197099, −4.83742244922653844135233795420, −4.76016505369932647820947080055, −4.72223964476221415109475682988, −4.38056870104588391142390359373, −4.00897604053955738836722710170, −3.85160082984493484948385184900, −3.83965080749752124721595933566, −3.75579654970022471664609909864, −3.41975297551013825950267292812, −3.34179194150733243741301246630, −2.99951132303552983284904893186, −2.71610249868806172257141623291, −2.43565002958516511346958203896, −2.28987874419423450495269966416, −2.25194070438566794164048200024, −2.02333429146801692597178282020, −1.59060936046664083177389947721, −1.29255634186707623467288052427, −1.09441886584919454489348562437, −1.08561613217301888949976881122, −0.39250421923980820701298727275, −0.19290917447958255108560666359, 0.19290917447958255108560666359, 0.39250421923980820701298727275, 1.08561613217301888949976881122, 1.09441886584919454489348562437, 1.29255634186707623467288052427, 1.59060936046664083177389947721, 2.02333429146801692597178282020, 2.25194070438566794164048200024, 2.28987874419423450495269966416, 2.43565002958516511346958203896, 2.71610249868806172257141623291, 2.99951132303552983284904893186, 3.34179194150733243741301246630, 3.41975297551013825950267292812, 3.75579654970022471664609909864, 3.83965080749752124721595933566, 3.85160082984493484948385184900, 4.00897604053955738836722710170, 4.38056870104588391142390359373, 4.72223964476221415109475682988, 4.76016505369932647820947080055, 4.83742244922653844135233795420, 5.14229331087035426504742197099, 5.43373247788282536580030267928, 5.57102585267548162113678575558

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