Properties

Label 8-74e4-1.1-c1e4-0-2
Degree $8$
Conductor $29986576$
Sign $1$
Analytic cond. $0.121908$
Root an. cond. $0.768695$
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  − 2·3-s + 4-s + 6·5-s + 4·7-s + 4·9-s − 12·11-s − 2·12-s − 12·13-s − 12·15-s + 6·17-s − 6·19-s + 6·20-s − 8·21-s + 11·25-s − 4·27-s + 4·28-s + 24·33-s + 24·35-s + 4·36-s − 2·37-s + 24·39-s − 6·41-s − 12·44-s + 24·45-s + 12·47-s + 18·49-s − 12·51-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 2.68·5-s + 1.51·7-s + 4/3·9-s − 3.61·11-s − 0.577·12-s − 3.32·13-s − 3.09·15-s + 1.45·17-s − 1.37·19-s + 1.34·20-s − 1.74·21-s + 11/5·25-s − 0.769·27-s + 0.755·28-s + 4.17·33-s + 4.05·35-s + 2/3·36-s − 0.328·37-s + 3.84·39-s − 0.937·41-s − 1.80·44-s + 3.57·45-s + 1.75·47-s + 18/7·49-s − 1.68·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(29986576\)    =    \(2^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(0.121908\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 29986576,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7828512011\)
\(L(\frac12)\) \(\approx\) \(0.7828512011\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
good3$D_4\times C_2$ \( 1 + 2 T - 4 T^{3} - 5 T^{4} - 4 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2^2$ \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 6 T + 13 T^{2} - 6 T^{3} - 84 T^{4} - 6 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 6 T + 44 T^{2} + 192 T^{3} + 891 T^{4} + 192 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 68 T^{2} + 2106 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 17 T^{2} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 100 T^{2} + 4314 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 + 6 T - 7 T^{2} - 234 T^{3} - 1308 T^{4} - 234 p T^{5} - 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 4 T^{2} - 3210 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 12 T + 14 T^{2} + 288 T^{3} + 6459 T^{4} + 288 p T^{5} + 14 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 12 T + 142 T^{2} + 1128 T^{3} + 8187 T^{4} + 1128 p T^{5} + 142 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 10 T - 32 T^{2} + 20 T^{3} + 6235 T^{4} + 20 p T^{5} - 32 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^3$ \( 1 - 130 T^{2} + 11859 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
79$D_4\times C_2$ \( 1 - 6 T - 52 T^{2} + 384 T^{3} - 1197 T^{4} + 384 p T^{5} - 52 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 6 T - 64 T^{2} - 396 T^{3} + 123 T^{4} - 396 p T^{5} - 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 18 T + 169 T^{2} - 1098 T^{3} + 5412 T^{4} - 1098 p T^{5} + 169 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 310 T^{2} + 42411 T^{4} - 310 p^{2} T^{6} + p^{4} T^{8} \)
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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

−10.72979023045933942202522845319, −10.34910272065349020369379331293, −10.22748427003330092383453510609, −10.22572874941855798934882086664, −9.834822439316199948385952778723, −9.687449684183083915423721605071, −9.273668859792150488751516843853, −8.757677978369228472229554727719, −8.287451503679754829241604697852, −7.892931623663597501548689123524, −7.57989835563077901132931397111, −7.48207459630864770676611344149, −7.28688928546112566849819576227, −6.61877935023988811205840373381, −6.20786681146086478404619554734, −5.85627710721084722349181152484, −5.52795291595256257150481002472, −5.24088143535986636179282696238, −4.99630177260622274496114811717, −4.89944297394910492360154880536, −4.46680475983487499123560289942, −3.21896739138208407882888504935, −2.27892757616353307373626938367, −2.19017534756175165655994352901, −2.17634053710953356201098501685, 2.17634053710953356201098501685, 2.19017534756175165655994352901, 2.27892757616353307373626938367, 3.21896739138208407882888504935, 4.46680475983487499123560289942, 4.89944297394910492360154880536, 4.99630177260622274496114811717, 5.24088143535986636179282696238, 5.52795291595256257150481002472, 5.85627710721084722349181152484, 6.20786681146086478404619554734, 6.61877935023988811205840373381, 7.28688928546112566849819576227, 7.48207459630864770676611344149, 7.57989835563077901132931397111, 7.892931623663597501548689123524, 8.287451503679754829241604697852, 8.757677978369228472229554727719, 9.273668859792150488751516843853, 9.687449684183083915423721605071, 9.834822439316199948385952778723, 10.22572874941855798934882086664, 10.22748427003330092383453510609, 10.34910272065349020369379331293, 10.72979023045933942202522845319

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