Properties

Label 8-760e4-1.1-c1e4-0-7
Degree $8$
Conductor $333621760000$
Sign $1$
Analytic cond. $1356.32$
Root an. cond. $2.46345$
Motivic weight $1$
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  + 6·9-s + 4·19-s + 8·25-s − 4·29-s + 8·31-s + 22·49-s − 52·59-s + 8·61-s − 16·71-s − 8·79-s + 17·81-s + 32·89-s + 24·101-s + 20·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + 24·171-s + 173-s + ⋯
L(s)  = 1  + 2·9-s + 0.917·19-s + 8/5·25-s − 0.742·29-s + 1.43·31-s + 22/7·49-s − 6.76·59-s + 1.02·61-s − 1.89·71-s − 0.900·79-s + 17/9·81-s + 3.39·89-s + 2.38·101-s + 1.91·109-s − 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.61·169-s + 1.83·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1356.32\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{760} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.476597357\)
\(L(\frac12)\) \(\approx\) \(4.476597357\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
19$C_1$ \( ( 1 - T )^{4} \)
good3$D_4\times C_2$ \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 22 T^{2} + 211 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 34 T^{2} + 595 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 54 T^{2} + 1715 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 4 T + 64 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 4 T^{2} + 694 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 128 T^{2} + 7506 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 194 T^{2} + 14995 T^{4} - 194 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 26 T + 285 T^{2} + 26 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 4 T + 124 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 166 T^{2} + 15667 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 34 T^{2} + 7075 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 4 T + 154 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 116 T^{2} + 6774 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 16 T + 224 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 316 T^{2} + 43270 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.36998093511099392059071773604, −7.21294797274446815649898991160, −6.98409059744952931501359193110, −6.95903627960470791402390147268, −6.56959323799765553176494737363, −6.15065904734731538771193300256, −6.05263617049722485361153257463, −5.80412128667269593975812286444, −5.73070765141885938716729093325, −5.26672112923341384837127180925, −4.78275441402741845589119708664, −4.72904715996149056883108191006, −4.58791245096905576973691042751, −4.30440458851706631729758103221, −4.26823320588900387482720975867, −3.68108591139365277382034526585, −3.23824469409920410147123240171, −3.18406575996491588129824393308, −3.10069969980703888622577382641, −2.50766382639485264894953352878, −2.02416744166037061161946241888, −1.85693303812387886420175814434, −1.42128487423258976484156251315, −0.972414182998217267453148965279, −0.66784810928231549694847414130, 0.66784810928231549694847414130, 0.972414182998217267453148965279, 1.42128487423258976484156251315, 1.85693303812387886420175814434, 2.02416744166037061161946241888, 2.50766382639485264894953352878, 3.10069969980703888622577382641, 3.18406575996491588129824393308, 3.23824469409920410147123240171, 3.68108591139365277382034526585, 4.26823320588900387482720975867, 4.30440458851706631729758103221, 4.58791245096905576973691042751, 4.72904715996149056883108191006, 4.78275441402741845589119708664, 5.26672112923341384837127180925, 5.73070765141885938716729093325, 5.80412128667269593975812286444, 6.05263617049722485361153257463, 6.15065904734731538771193300256, 6.56959323799765553176494737363, 6.95903627960470791402390147268, 6.98409059744952931501359193110, 7.21294797274446815649898991160, 7.36998093511099392059071773604

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