Properties

Label 8-70e4-1.1-c1e4-0-2
Degree $8$
Conductor $24010000$
Sign $1$
Analytic cond. $0.0976114$
Root an. cond. $0.747631$
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-s + 4·5-s − 6·9-s − 6·11-s − 10·19-s + 4·20-s + 5·25-s + 16·29-s + 4·31-s − 6·36-s + 12·41-s − 6·44-s − 24·45-s − 2·49-s − 24·55-s − 8·59-s − 12·61-s − 64-s − 24·71-s − 10·76-s + 28·79-s + 9·81-s + 4·89-s − 40·95-s + 36·99-s + 5·100-s + 4·109-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.78·5-s − 2·9-s − 1.80·11-s − 2.29·19-s + 0.894·20-s + 25-s + 2.97·29-s + 0.718·31-s − 36-s + 1.87·41-s − 0.904·44-s − 3.57·45-s − 2/7·49-s − 3.23·55-s − 1.04·59-s − 1.53·61-s − 1/8·64-s − 2.84·71-s − 1.14·76-s + 3.15·79-s + 81-s + 0.423·89-s − 4.10·95-s + 3.61·99-s + 1/2·100-s + 0.383·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(24010000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.0976114\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{70} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 24010000,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8037536565\)
\(L(\frac12)\) \(\approx\) \(0.8037536565\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
good3$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
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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

−11.00288960029022323344523879439, −10.36885151170820898870890211826, −10.33937039601442105215947901050, −10.25321854300404979655528034078, −10.15172993684019108314376753627, −9.257965064754940141499293867865, −9.130416190934001159258178329616, −8.806108920344191086922236346895, −8.732000280773706090483137127007, −7.963653473364360370948674694096, −7.903820222778613211112110032568, −7.897795448188747920263290077983, −7.08691500602834560351759685181, −6.42033470391414466067033464248, −6.37698392419049473313456733974, −6.18661944655002757176010237946, −5.86277139977059120087424334488, −5.31991896541835435014788028969, −5.17048229971270943599787253056, −4.48972306656910139104697797137, −4.32092512076614076650127781573, −3.04827988601823267908083379659, −2.84781537432164549982239978598, −2.50548909992439682869609315126, −1.98793399665740071309754400575, 1.98793399665740071309754400575, 2.50548909992439682869609315126, 2.84781537432164549982239978598, 3.04827988601823267908083379659, 4.32092512076614076650127781573, 4.48972306656910139104697797137, 5.17048229971270943599787253056, 5.31991896541835435014788028969, 5.86277139977059120087424334488, 6.18661944655002757176010237946, 6.37698392419049473313456733974, 6.42033470391414466067033464248, 7.08691500602834560351759685181, 7.897795448188747920263290077983, 7.903820222778613211112110032568, 7.963653473364360370948674694096, 8.732000280773706090483137127007, 8.806108920344191086922236346895, 9.130416190934001159258178329616, 9.257965064754940141499293867865, 10.15172993684019108314376753627, 10.25321854300404979655528034078, 10.33937039601442105215947901050, 10.36885151170820898870890211826, 11.00288960029022323344523879439

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