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Properties

Label 8-2645e4-1.1-c1e4-0-0
Degree $8$
Conductor $4.894\times 10^{13}$
Sign $1$
Analytic cond. $198980.$
Root an. cond. $4.59569$
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  − 2-s − 3-s + 4-s − 4·5-s + 6-s + 7·7-s − 2·8-s − 9-s + 4·10-s + 3·11-s − 12-s + 4·13-s − 7·14-s + 4·15-s − 16-s + 5·17-s + 18-s + 8·19-s − 4·20-s − 7·21-s − 3·22-s + 2·24-s + 10·25-s − 4·26-s + 27-s + 7·28-s + 3·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 2.64·7-s − 0.707·8-s − 1/3·9-s + 1.26·10-s + 0.904·11-s − 0.288·12-s + 1.10·13-s − 1.87·14-s + 1.03·15-s − 1/4·16-s + 1.21·17-s + 0.235·18-s + 1.83·19-s − 0.894·20-s − 1.52·21-s − 0.639·22-s + 0.408·24-s + 2·25-s − 0.784·26-s + 0.192·27-s + 1.32·28-s + 0.557·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(198980.\)
Root analytic conductor: \(4.59569\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 23^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.372315221\)
\(L(\frac12)\) \(\approx\) \(1.372315221\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$ \( ( 1 + T )^{4} \)
23 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + T + T^{3} + p^{2} T^{4} + p T^{5} + p^{3} T^{7} + p^{4} T^{8} \)
3$C_2 \wr S_4$ \( 1 + T + 2 T^{2} + 2 T^{3} + 2 p T^{4} + 2 p T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 - p T + 38 T^{2} - 20 p T^{3} + 432 T^{4} - 20 p^{2} T^{5} + 38 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 3 T + 9 T^{2} - 37 T^{3} + 247 T^{4} - 37 p T^{5} + 9 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 4 T + 29 T^{2} - 149 T^{3} + 438 T^{4} - 149 p T^{5} + 29 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 5 T + 22 T^{2} - 54 T^{3} + 332 T^{4} - 54 p T^{5} + 22 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 8 T + 67 T^{2} - 319 T^{3} + 1757 T^{4} - 319 p T^{5} + 67 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 3 T + 81 T^{2} - 10 p T^{3} + 3010 T^{4} - 10 p^{2} T^{5} + 81 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 5 T + 125 T^{2} + 451 T^{3} + 5829 T^{4} + 451 p T^{5} + 125 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 13 T + 152 T^{2} - 1002 T^{3} + 7214 T^{4} - 1002 p T^{5} + 152 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 8 T + 159 T^{2} + 947 T^{3} + 9673 T^{4} + 947 p T^{5} + 159 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 20 T + 223 T^{2} - 1779 T^{3} + 11916 T^{4} - 1779 p T^{5} + 223 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 11 T + 193 T^{2} + 1500 T^{3} + 13670 T^{4} + 1500 p T^{5} + 193 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 5 T + 194 T^{2} - 732 T^{3} + 15054 T^{4} - 732 p T^{5} + 194 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 4 T + 123 T^{2} + 511 T^{3} + 7768 T^{4} + 511 p T^{5} + 123 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 16 T + 242 T^{2} + 2552 T^{3} + 21579 T^{4} + 2552 p T^{5} + 242 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 34 T + 659 T^{2} - 8505 T^{3} + 80930 T^{4} - 8505 p T^{5} + 659 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 6 T + 21 T^{2} + 449 T^{3} + 9583 T^{4} + 449 p T^{5} + 21 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 19 T + 214 T^{2} + 1740 T^{3} + 15348 T^{4} + 1740 p T^{5} + 214 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 24 T + 398 T^{2} - 4888 T^{3} + 49799 T^{4} - 4888 p T^{5} + 398 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 10 T + 301 T^{2} + 2379 T^{3} + 36296 T^{4} + 2379 p T^{5} + 301 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 19 T + 338 T^{2} + 44 p T^{3} + 40764 T^{4} + 44 p^{2} T^{5} + 338 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 17 T + 488 T^{2} + 5182 T^{3} + 76038 T^{4} + 5182 p T^{5} + 488 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.13928874322895800469661488908, −6.11069581432233676397255145203, −5.97424157423493810998164060555, −5.58238238454728443747585297280, −5.48197524426819664139848482676, −5.26580511674169593515667697772, −4.97562293566546158250719947234, −4.86254792088028342255886752756, −4.62517435603900993352879837928, −4.46217577625141151790787608470, −4.22067205716089277557858510833, −3.92631017820146989378480270368, −3.80639721507656887006574330327, −3.45646260349543633369305228435, −3.32207815479823904231050182981, −3.17528732574420655490716472518, −2.66043404374401836823921783318, −2.48652523314658017706917366827, −2.36829879704113899686428824981, −1.68175797809050161581907311556, −1.50500180255675255383512152276, −1.27713955410095055806148271155, −1.05765368497862259260253892632, −0.859964896923663688195269238622, −0.23978708921957479883887784040, 0.23978708921957479883887784040, 0.859964896923663688195269238622, 1.05765368497862259260253892632, 1.27713955410095055806148271155, 1.50500180255675255383512152276, 1.68175797809050161581907311556, 2.36829879704113899686428824981, 2.48652523314658017706917366827, 2.66043404374401836823921783318, 3.17528732574420655490716472518, 3.32207815479823904231050182981, 3.45646260349543633369305228435, 3.80639721507656887006574330327, 3.92631017820146989378480270368, 4.22067205716089277557858510833, 4.46217577625141151790787608470, 4.62517435603900993352879837928, 4.86254792088028342255886752756, 4.97562293566546158250719947234, 5.26580511674169593515667697772, 5.48197524426819664139848482676, 5.58238238454728443747585297280, 5.97424157423493810998164060555, 6.11069581432233676397255145203, 6.13928874322895800469661488908

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