Properties

Label 8-980e4-1.1-c1e4-0-10
Degree $8$
Conductor $922368160000$
Sign $1$
Analytic cond. $3749.83$
Root an. cond. $2.79738$
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  − 4·5-s + 3·9-s − 6·11-s + 16·19-s + 5·25-s + 4·29-s − 4·31-s + 24·41-s − 12·45-s + 24·55-s + 20·59-s − 14·79-s + 9·81-s − 16·89-s − 64·95-s − 18·99-s − 24·101-s − 14·109-s + 31·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.78·5-s + 9-s − 1.80·11-s + 3.67·19-s + 25-s + 0.742·29-s − 0.718·31-s + 3.74·41-s − 1.78·45-s + 3.23·55-s + 2.60·59-s − 1.57·79-s + 81-s − 1.69·89-s − 6.56·95-s − 1.80·99-s − 2.38·101-s − 1.34·109-s + 2.81·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3749.83\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{980} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.962611822\)
\(L(\frac12)\) \(\approx\) \(1.962611822\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good3$C_2$$\times$$C_2^2$ \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \)
11$C_2^2$ \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 9 T^{2} - 208 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \)
23$C_2^3$ \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
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$ \( ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 27 T^{2} - 1480 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 185 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.31334669445838623682639990113, −7.10365631757318600519899018152, −7.00048018237222261760213757137, −6.54246563685918802109861910950, −6.38811216024287723597101720380, −5.69432002156047575343929288719, −5.67800944691663403298356849737, −5.59450076491160364774826823158, −5.52920260136622659280937084706, −5.02270565418853954589469780708, −4.78820270560852647865282149702, −4.66841974426817634879557763709, −4.17458753199840679756951925377, −4.15573363442531655335855105067, −3.82845209086311144048413634782, −3.71608156313355205832902244823, −3.16506432793313423237778978685, −3.04920154635995798618464664738, −2.72016682766894819974051636666, −2.67915935985273201602604406358, −2.10716521083837070371705689821, −1.68824620037295453690926620688, −1.02484058119904222543029692529, −0.966976187334345446841872998318, −0.42636961076067710366381340074, 0.42636961076067710366381340074, 0.966976187334345446841872998318, 1.02484058119904222543029692529, 1.68824620037295453690926620688, 2.10716521083837070371705689821, 2.67915935985273201602604406358, 2.72016682766894819974051636666, 3.04920154635995798618464664738, 3.16506432793313423237778978685, 3.71608156313355205832902244823, 3.82845209086311144048413634782, 4.15573363442531655335855105067, 4.17458753199840679756951925377, 4.66841974426817634879557763709, 4.78820270560852647865282149702, 5.02270565418853954589469780708, 5.52920260136622659280937084706, 5.59450076491160364774826823158, 5.67800944691663403298356849737, 5.69432002156047575343929288719, 6.38811216024287723597101720380, 6.54246563685918802109861910950, 7.00048018237222261760213757137, 7.10365631757318600519899018152, 7.31334669445838623682639990113

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