Properties

Label 8-1520e4-1.1-c1e4-0-6
Degree $8$
Conductor $5.338\times 10^{12}$
Sign $1$
Analytic cond. $21701.1$
Root an. cond. $3.48385$
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  + 6·9-s + 12·11-s − 4·19-s + 10·25-s + 16·31-s − 24·41-s + 12·49-s − 24·59-s − 20·61-s + 24·71-s + 8·79-s + 14·81-s + 24·89-s + 72·99-s + 12·101-s − 8·109-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + ⋯
L(s)  = 1  + 2·9-s + 3.61·11-s − 0.917·19-s + 2·25-s + 2.87·31-s − 3.74·41-s + 12/7·49-s − 3.12·59-s − 2.56·61-s + 2.84·71-s + 0.900·79-s + 14/9·81-s + 2.54·89-s + 7.23·99-s + 1.19·101-s − 0.766·109-s + 5.09·121-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 + 6/13·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(21701.1\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1520} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.884056498\)
\(L(\frac12)\) \(\approx\) \(7.884056498\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_1$ \( ( 1 + T )^{4} \)
good3$D_4\times C_2$ \( 1 - 2 p T^{2} + 22 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \)
7$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
11$C_4$ \( ( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 6 T^{2} + 302 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 12 T^{2} - 106 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 36 T^{2} + 662 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 94 T^{2} + 4542 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 158 T^{2} + 11454 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 142 T^{2} + 10374 T^{4} - 142 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 76 T^{2} + 5622 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 116 T^{2} + 10662 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
89$C_4$ \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 174 T^{2} + 18782 T^{4} - 174 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.73338720964366953546835899141, −6.59405542835070585831169088581, −6.58398322769346663071247319122, −6.21733726424002316977766055676, −6.08678966076541757880115864477, −5.63888996969578368688672453369, −5.62069275659300010840208802164, −4.98178539175047273138265417363, −4.69478788221482684592070565224, −4.64367850574787660840022587846, −4.60092908589301462952784790211, −4.55380762098603427637189860559, −3.93149059235761268457382891037, −3.92299089664035399741985989539, −3.57236117439916879968614843310, −3.37792996068683155764169322055, −3.20096910062222297460504201692, −2.91104047446450596028450771723, −2.26374560378313040173124509091, −2.19174949254656807452003272147, −1.77219049009842438974043788465, −1.35922589834201333082967658028, −1.27329487305174095836253820266, −1.08853593004070106684044245866, −0.54542025728583361434968334480, 0.54542025728583361434968334480, 1.08853593004070106684044245866, 1.27329487305174095836253820266, 1.35922589834201333082967658028, 1.77219049009842438974043788465, 2.19174949254656807452003272147, 2.26374560378313040173124509091, 2.91104047446450596028450771723, 3.20096910062222297460504201692, 3.37792996068683155764169322055, 3.57236117439916879968614843310, 3.92299089664035399741985989539, 3.93149059235761268457382891037, 4.55380762098603427637189860559, 4.60092908589301462952784790211, 4.64367850574787660840022587846, 4.69478788221482684592070565224, 4.98178539175047273138265417363, 5.62069275659300010840208802164, 5.63888996969578368688672453369, 6.08678966076541757880115864477, 6.21733726424002316977766055676, 6.58398322769346663071247319122, 6.59405542835070585831169088581, 6.73338720964366953546835899141

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