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Properties

Label 8-4002e4-1.1-c1e4-0-0
Degree $8$
Conductor $2.565\times 10^{14}$
Sign $1$
Analytic cond. $1.04283\times 10^{6}$
Root an. cond. $5.65297$
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  − 4·2-s − 4·3-s + 10·4-s + 2·5-s + 16·6-s + 6·7-s − 20·8-s + 10·9-s − 8·10-s − 40·12-s − 24·14-s − 8·15-s + 35·16-s − 6·17-s − 40·18-s + 2·19-s + 20·20-s − 24·21-s + 4·23-s + 80·24-s − 5·25-s − 20·27-s + 60·28-s − 4·29-s + 32·30-s + 8·31-s − 56·32-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s + 0.894·5-s + 6.53·6-s + 2.26·7-s − 7.07·8-s + 10/3·9-s − 2.52·10-s − 11.5·12-s − 6.41·14-s − 2.06·15-s + 35/4·16-s − 1.45·17-s − 9.42·18-s + 0.458·19-s + 4.47·20-s − 5.23·21-s + 0.834·23-s + 16.3·24-s − 25-s − 3.84·27-s + 11.3·28-s − 0.742·29-s + 5.84·30-s + 1.43·31-s − 9.89·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 23^{4} \cdot 29^{4}\)
Sign: $1$
Analytic conductor: \(1.04283\times 10^{6}\)
Root analytic conductor: \(5.65297\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 23^{4} \cdot 29^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.455317776\)
\(L(\frac12)\) \(\approx\) \(1.455317776\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
3$C_1$ \( ( 1 + T )^{4} \)
23$C_1$ \( ( 1 - T )^{4} \)
29$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + 36 T^{4} - 2 p^{2} T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 29 T^{2} - 94 T^{3} + 276 T^{4} - 94 p T^{5} + 29 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 + 18 T^{2} - 24 T^{3} + 162 T^{4} - 24 p T^{5} + 18 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 10 T^{2} + 40 T^{3} - 70 T^{4} + 40 p T^{5} + 10 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 37 T^{2} + 162 T^{3} + 736 T^{4} + 162 p T^{5} + 37 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 17 T^{2} - 54 T^{3} + 696 T^{4} - 54 p T^{5} + 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 122 T^{2} - 648 T^{3} + 5514 T^{4} - 648 p T^{5} + 122 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 59 T^{2} - 102 T^{3} + 60 T^{4} - 102 p T^{5} + 59 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 143 T^{2} - 694 T^{3} + 8360 T^{4} - 694 p T^{5} + 143 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 201 T^{2} - 1690 T^{3} + 13464 T^{4} - 1690 p T^{5} + 201 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 213 T^{2} - 1418 T^{3} + 15588 T^{4} - 1418 p T^{5} + 213 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 112 T^{2} - 956 T^{3} + 5998 T^{4} - 956 p T^{5} + 112 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 373 T^{2} + 4094 T^{3} + 36940 T^{4} + 4094 p T^{5} + 373 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 28 T + 432 T^{2} - 4484 T^{3} + 38094 T^{4} - 4484 p T^{5} + 432 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 332 T^{2} - 3608 T^{3} + 32886 T^{4} - 3608 p T^{5} + 332 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 28 T + 462 T^{2} - 5068 T^{3} + 46754 T^{4} - 5068 p T^{5} + 462 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 92 T^{2} - 512 T^{3} + 3558 T^{4} - 512 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 214 T^{2} - 2500 T^{3} + 23066 T^{4} - 2500 p T^{5} + 214 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 408 T^{2} - 4916 T^{3} + 53726 T^{4} - 4916 p T^{5} + 408 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 332 T^{2} + 2632 T^{3} + 21878 T^{4} + 2632 p T^{5} + 332 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 32 T + 700 T^{2} + 10272 T^{3} + 117878 T^{4} + 10272 p T^{5} + 700 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.28636572405820789937375183650, −5.57847806407679861399926157271, −5.56173003943525570699708667975, −5.51874732136206612015158149986, −5.50863859880605912001720531561, −5.05008325526529854422952873931, −5.00066059664194345080164445834, −4.70674940525600118228113621584, −4.43610033059010684345837261443, −4.14820613937935349855855121250, −4.06031899064603669674149599849, −3.79117649733069022940862403101, −3.73190445303366159665508059577, −2.98834407728878027897381962259, −2.68426144999371300567791138423, −2.63980689308710913991800128661, −2.38882215644980644418351265159, −2.01502943840024056540401026618, −1.94125957294907385844774291203, −1.64627094991400125478324745277, −1.49845518981383636821181098975, −1.02653446190447595663648990950, −0.75341244899967336848972577795, −0.65810877460570271866895200525, −0.51000000635580593785659963192, 0.51000000635580593785659963192, 0.65810877460570271866895200525, 0.75341244899967336848972577795, 1.02653446190447595663648990950, 1.49845518981383636821181098975, 1.64627094991400125478324745277, 1.94125957294907385844774291203, 2.01502943840024056540401026618, 2.38882215644980644418351265159, 2.63980689308710913991800128661, 2.68426144999371300567791138423, 2.98834407728878027897381962259, 3.73190445303366159665508059577, 3.79117649733069022940862403101, 4.06031899064603669674149599849, 4.14820613937935349855855121250, 4.43610033059010684345837261443, 4.70674940525600118228113621584, 5.00066059664194345080164445834, 5.05008325526529854422952873931, 5.50863859880605912001720531561, 5.51874732136206612015158149986, 5.56173003943525570699708667975, 5.57847806407679861399926157271, 6.28636572405820789937375183650

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