Properties

Label 8-285e4-1.1-c1e4-0-0
Degree $8$
Conductor $6597500625$
Sign $1$
Analytic cond. $26.8217$
Root an. cond. $1.50855$
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  − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 4·7-s − 2·8-s + 9-s − 4·10-s + 6·12-s − 2·13-s + 8·14-s + 4·15-s + 8·17-s − 2·18-s + 16·19-s + 6·20-s − 8·21-s − 4·23-s − 4·24-s + 25-s + 4·26-s − 2·27-s − 12·28-s − 8·29-s − 8·30-s − 20·31-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 1.51·7-s − 0.707·8-s + 1/3·9-s − 1.26·10-s + 1.73·12-s − 0.554·13-s + 2.13·14-s + 1.03·15-s + 1.94·17-s − 0.471·18-s + 3.67·19-s + 1.34·20-s − 1.74·21-s − 0.834·23-s − 0.816·24-s + 1/5·25-s + 0.784·26-s − 0.384·27-s − 2.26·28-s − 1.48·29-s − 1.46·30-s − 3.59·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(26.8217\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.301225439\)
\(L(\frac12)\) \(\approx\) \(1.301225439\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( ( 1 - T + T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 2 T - 15 T^{2} - 14 T^{3} + 140 T^{4} - 14 p T^{5} - 15 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 8 T + 22 T^{2} - 64 T^{3} + 387 T^{4} - 64 p T^{5} + 22 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_4\times C_2$ \( 1 + 4 T - 26 T^{2} - 16 T^{3} + 867 T^{4} - 16 p T^{5} - 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 8 T + 22 T^{2} - 128 T^{3} - 933 T^{4} - 128 p T^{5} + 22 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
37$D_{4}$ \( ( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^3$ \( 1 - 74 T^{2} + 3795 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 10 T - 3 T^{2} + 170 T^{3} + 4460 T^{4} + 170 p T^{5} - 3 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 12 T + 22 T^{2} - 336 T^{3} + 5907 T^{4} - 336 p T^{5} + 22 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 6 T + 33 T^{2} - 714 T^{3} - 5908 T^{4} - 714 p T^{5} + 33 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 6 T - 35 T^{2} + 378 T^{3} - 1860 T^{4} + 378 p T^{5} - 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 10 T + 29 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 2 T - 71 T^{2} - 142 T^{3} + 4 T^{4} - 142 p T^{5} - 71 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 18 T + 117 T^{2} - 882 T^{3} + 11012 T^{4} - 882 p T^{5} + 117 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
89$D_4\times C_2$ \( 1 + 8 T - 58 T^{2} - 448 T^{3} + 1267 T^{4} - 448 p T^{5} - 58 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−8.697558677410357641512976589855, −8.624376751737401864417724159078, −7.918550472571814294428617188433, −7.83511083891169464320067762999, −7.67873442247091151783772898953, −7.25369428120164197946998489691, −7.16736792364550433085930013339, −7.03783851096672845128185869822, −7.02607939682509670102950005724, −6.13782312693259007017630299683, −5.93176973212319772889184512126, −5.71716655994584507347787726499, −5.61132187096502854683716130706, −5.18213437904242541020277990687, −5.08596275089432113087427880404, −4.43315563257410613202038051105, −3.83022655100526189477527971678, −3.38309768630268570955118127202, −3.33620189312162287666657318407, −3.13314731433638830095454856736, −3.00517892019937582283499914898, −1.85927664038646741948367273824, −1.83446124223657995218546554359, −1.77082502184961863227353430648, −0.62688325020116033969017506592, 0.62688325020116033969017506592, 1.77082502184961863227353430648, 1.83446124223657995218546554359, 1.85927664038646741948367273824, 3.00517892019937582283499914898, 3.13314731433638830095454856736, 3.33620189312162287666657318407, 3.38309768630268570955118127202, 3.83022655100526189477527971678, 4.43315563257410613202038051105, 5.08596275089432113087427880404, 5.18213437904242541020277990687, 5.61132187096502854683716130706, 5.71716655994584507347787726499, 5.93176973212319772889184512126, 6.13782312693259007017630299683, 7.02607939682509670102950005724, 7.03783851096672845128185869822, 7.16736792364550433085930013339, 7.25369428120164197946998489691, 7.67873442247091151783772898953, 7.83511083891169464320067762999, 7.918550472571814294428617188433, 8.624376751737401864417724159078, 8.697558677410357641512976589855

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