Properties

Label 8-35e8-1.1-c1e4-0-4
Degree $8$
Conductor $2.252\times 10^{12}$
Sign $1$
Analytic cond. $9154.88$
Root an. cond. $3.12756$
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  + 4·4-s + 6·9-s − 12·11-s + 4·16-s + 24·19-s + 12·29-s − 24·31-s + 24·36-s + 8·41-s − 48·44-s + 8·59-s − 16·64-s − 24·71-s + 96·76-s + 28·79-s + 17·81-s − 32·89-s − 72·99-s + 24·101-s − 20·109-s + 48·116-s + 62·121-s − 96·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 2·4-s + 2·9-s − 3.61·11-s + 16-s + 5.50·19-s + 2.22·29-s − 4.31·31-s + 4·36-s + 1.24·41-s − 7.23·44-s + 1.04·59-s − 2·64-s − 2.84·71-s + 11.0·76-s + 3.15·79-s + 17/9·81-s − 3.39·89-s − 7.23·99-s + 2.38·101-s − 1.91·109-s + 4.45·116-s + 5.63·121-s − 8.62·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(9154.88\)
Root analytic conductor: \(3.12756\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.522267366\)
\(L(\frac12)\) \(\approx\) \(6.522267366\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
3$D_4\times C_2$ \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 30 T^{2} + 491 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 30 T^{2} + 43 p T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
23$D_4\times C_2$ \( 1 - 16 T^{2} + 834 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 104 T^{2} + 5154 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 134 T^{2} + 8259 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 88 T^{2} + p^{2} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 - 4 T + 104 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 200 T^{2} + 17826 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \)
71$C_4$ \( ( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{4} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
97$D_4\times C_2$ \( 1 - 190 T^{2} + 22011 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.98796468974389502323412828151, −6.98363290492746593740983918365, −6.84045061439501602714711681239, −6.34696800522362573593715694532, −5.84192138597272278981041339639, −5.78928336388142164310111430713, −5.64471692916712948508154018816, −5.33421194560388637154149162104, −5.32645511438736651570010808757, −4.97473790733920081220180753826, −4.87825936341457816741149423533, −4.53872707093786199692961572561, −4.10721730909854452609510069331, −4.02768448642237215175657012170, −3.43116280685049105229478629439, −3.25529125406817869078892988610, −3.00045444112875943221989240551, −2.94969784261575023663820918899, −2.68184558177200711914750519250, −2.28259666147370226184679726090, −2.01953311062371501653911428773, −1.65146466393061523194375995608, −1.44714350332016053173373185346, −0.959370353630057669573196307846, −0.50832121335449841800534236296, 0.50832121335449841800534236296, 0.959370353630057669573196307846, 1.44714350332016053173373185346, 1.65146466393061523194375995608, 2.01953311062371501653911428773, 2.28259666147370226184679726090, 2.68184558177200711914750519250, 2.94969784261575023663820918899, 3.00045444112875943221989240551, 3.25529125406817869078892988610, 3.43116280685049105229478629439, 4.02768448642237215175657012170, 4.10721730909854452609510069331, 4.53872707093786199692961572561, 4.87825936341457816741149423533, 4.97473790733920081220180753826, 5.32645511438736651570010808757, 5.33421194560388637154149162104, 5.64471692916712948508154018816, 5.78928336388142164310111430713, 5.84192138597272278981041339639, 6.34696800522362573593715694532, 6.84045061439501602714711681239, 6.98363290492746593740983918365, 6.98796468974389502323412828151

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