Properties

Label 8-2352e4-1.1-c2e4-0-8
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $1.68690\times 10^{7}$
Root an. cond. $8.00545$
Motivic weight $2$
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  − 4·5-s − 6·9-s − 32·13-s + 52·17-s − 48·25-s − 8·29-s + 80·37-s − 68·41-s + 24·45-s + 16·53-s + 264·61-s + 128·65-s + 272·73-s + 27·81-s − 208·85-s + 172·89-s − 320·97-s + 436·101-s − 88·109-s + 424·113-s + 192·117-s + 464·121-s + 228·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 4/5·5-s − 2/3·9-s − 2.46·13-s + 3.05·17-s − 1.91·25-s − 0.275·29-s + 2.16·37-s − 1.65·41-s + 8/15·45-s + 0.301·53-s + 4.32·61-s + 1.96·65-s + 3.72·73-s + 1/3·81-s − 2.44·85-s + 1.93·89-s − 3.29·97-s + 4.31·101-s − 0.807·109-s + 3.75·113-s + 1.64·117-s + 3.83·121-s + 1.82·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.68690\times 10^{7}\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.364433053\)
\(L(\frac12)\) \(\approx\) \(5.364433053\)
\(L(2)\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( ( 1 + 2 T + 6 p T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 464 T^{2} + 83022 T^{4} - 464 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 + 16 T + 318 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 26 T + 558 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 1124 T^{2} + 554982 T^{4} - 1124 p^{4} T^{6} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 1280 T^{2} + 866382 T^{4} - 1280 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{4} \)
31$D_4\times C_2$ \( 1 - 1444 T^{2} + 1594182 T^{4} - 1444 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 - 40 T + 2382 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 34 T + 102 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 2276 T^{2} + 2627622 T^{4} - 2276 p^{4} T^{6} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 - 7156 T^{2} + 21968742 T^{4} - 7156 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 - 8 T + 3534 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 5204 T^{2} + 17832582 T^{4} - 5204 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 - 132 T + 8774 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 12244 T^{2} + 72446790 T^{4} - 12244 p^{4} T^{6} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 12704 T^{2} + 90771342 T^{4} - 12704 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 136 T + 14526 T^{2} - 136 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 11764 T^{2} + 106099302 T^{4} - 11764 p^{4} T^{6} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 13124 T^{2} + 100043430 T^{4} - 13124 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 - 86 T + 11622 T^{2} - 86 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 160 T + 24462 T^{2} + 160 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−6.24666005125534953115223614269, −5.81864141987371806234379322813, −5.74077286955525622593071484470, −5.53235187102480080306792410962, −5.48986672080343787235859295435, −4.98270140691602400346375030752, −4.93125317905845821820078620473, −4.85731647006165248382884759915, −4.73421578266227239571332649285, −4.00906684809652610749078244715, −3.96537874296615215659723726840, −3.87338681561528541604838307060, −3.74239688550247573896395756395, −3.30423873242890170150474385739, −3.11450493761426200643419194707, −2.88021281751218388185183933981, −2.79287603186568256330264325231, −2.20544564036786792349962734169, −2.11968399458247026660151130961, −1.87471093353470660055538753674, −1.78520858896390217917524749458, −0.854620506712297607983094383055, −0.841891824488564857182415727780, −0.48659485308720544301425632875, −0.46071544798692331345826567640, 0.46071544798692331345826567640, 0.48659485308720544301425632875, 0.841891824488564857182415727780, 0.854620506712297607983094383055, 1.78520858896390217917524749458, 1.87471093353470660055538753674, 2.11968399458247026660151130961, 2.20544564036786792349962734169, 2.79287603186568256330264325231, 2.88021281751218388185183933981, 3.11450493761426200643419194707, 3.30423873242890170150474385739, 3.74239688550247573896395756395, 3.87338681561528541604838307060, 3.96537874296615215659723726840, 4.00906684809652610749078244715, 4.73421578266227239571332649285, 4.85731647006165248382884759915, 4.93125317905845821820078620473, 4.98270140691602400346375030752, 5.48986672080343787235859295435, 5.53235187102480080306792410962, 5.74077286955525622593071484470, 5.81864141987371806234379322813, 6.24666005125534953115223614269

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