Properties

Label 8-315e4-1.1-c1e4-0-16
Degree $8$
Conductor $9845600625$
Sign $1$
Analytic cond. $40.0267$
Root an. cond. $1.58596$
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 + 4·5-s + 4·7-s + 7·16-s + 16·20-s + 10·25-s + 16·28-s + 16·35-s + 8·37-s + 8·43-s − 24·47-s − 2·49-s − 24·59-s + 8·64-s − 16·67-s − 16·79-s + 28·80-s + 24·83-s − 24·89-s + 40·100-s + 32·109-s + 28·112-s + 16·121-s + 20·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 2·4-s + 1.78·5-s + 1.51·7-s + 7/4·16-s + 3.57·20-s + 2·25-s + 3.02·28-s + 2.70·35-s + 1.31·37-s + 1.21·43-s − 3.50·47-s − 2/7·49-s − 3.12·59-s + 64-s − 1.95·67-s − 1.80·79-s + 3.13·80-s + 2.63·83-s − 2.54·89-s + 4·100-s + 3.06·109-s + 2.64·112-s + 1.45·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(40.0267\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.201869001\)
\(L(\frac12)\) \(\approx\) \(6.201869001\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 - p^{2} T^{2} + 9 T^{4} - p^{4} T^{6} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 16 T^{2} + 114 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 4 T^{2} - 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 28 T^{2} + 486 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 76 T^{2} + 2934 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 88 T^{2} + 5826 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 52 T^{2} + 1206 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + 32 T^{2} + 78 p T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 244 T^{2} + 25110 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 52 T^{2} + 15606 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
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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

−8.675618306953813845613530024764, −7.906440888919063439269924225338, −7.80977471373673174662320924812, −7.79442081431534446587376304985, −7.69874870166444998219624472639, −7.02810906694292914505111306980, −6.88171267493195176981269538222, −6.62681788617958233155747121512, −6.50067855380537693524795530447, −5.99628308600570897415193958069, −5.93738981558368653400554332536, −5.84476632100703426418965401639, −5.39477166136413814713265343408, −4.97165306204623213123493477573, −4.76529562969078925914512920827, −4.48060314309832582078767236639, −4.36218637575472330331694073788, −3.52459565625723560950418041350, −3.20171324690582010984728758088, −2.78724457082843949553297470231, −2.77003279710001407088362527445, −2.02453488071682227673094325807, −1.85199912483380229660003245258, −1.65645164193364391998557010203, −1.20480641101168406966098597349, 1.20480641101168406966098597349, 1.65645164193364391998557010203, 1.85199912483380229660003245258, 2.02453488071682227673094325807, 2.77003279710001407088362527445, 2.78724457082843949553297470231, 3.20171324690582010984728758088, 3.52459565625723560950418041350, 4.36218637575472330331694073788, 4.48060314309832582078767236639, 4.76529562969078925914512920827, 4.97165306204623213123493477573, 5.39477166136413814713265343408, 5.84476632100703426418965401639, 5.93738981558368653400554332536, 5.99628308600570897415193958069, 6.50067855380537693524795530447, 6.62681788617958233155747121512, 6.88171267493195176981269538222, 7.02810906694292914505111306980, 7.69874870166444998219624472639, 7.79442081431534446587376304985, 7.80977471373673174662320924812, 7.906440888919063439269924225338, 8.675618306953813845613530024764

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