Properties

Label 4-8036e2-1.1-c1e2-0-0
Degree $4$
Conductor $64577296$
Sign $1$
Analytic cond. $4117.50$
Root an. cond. $8.01047$
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  + 3·3-s + 3·5-s + 4·9-s + 2·11-s + 10·13-s + 9·15-s + 2·17-s + 3·19-s − 3·23-s + 6·27-s + 17·29-s − 11·31-s + 6·33-s + 6·37-s + 30·39-s + 2·41-s − 16·43-s + 12·45-s + 18·47-s + 6·51-s − 9·53-s + 6·55-s + 9·57-s − 59-s + 6·61-s + 30·65-s − 7·67-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.34·5-s + 4/3·9-s + 0.603·11-s + 2.77·13-s + 2.32·15-s + 0.485·17-s + 0.688·19-s − 0.625·23-s + 1.15·27-s + 3.15·29-s − 1.97·31-s + 1.04·33-s + 0.986·37-s + 4.80·39-s + 0.312·41-s − 2.43·43-s + 1.78·45-s + 2.62·47-s + 0.840·51-s − 1.23·53-s + 0.809·55-s + 1.19·57-s − 0.130·59-s + 0.768·61-s + 3.72·65-s − 0.855·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64577296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64577296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(64577296\)    =    \(2^{4} \cdot 7^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(4117.50\)
Root analytic conductor: \(8.01047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 64577296,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(13.65970371\)
\(L(\frac12)\) \(\approx\) \(13.65970371\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
41$C_1$ \( ( 1 - T )^{2} \)
good3$C_4$ \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 2 T - p T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T + 45 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 17 T + 127 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 11 T + 63 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 16 T + 137 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 9 T + 123 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T + 89 T^{2} + p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 7 T + 117 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 10 T + 115 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 26 T + 314 T^{2} + 26 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + T + 97 T^{2} + p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 7 T + 47 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.188321110831117079718253589436, −7.71638990036421097168800748589, −7.36320937280697382074233550496, −7.09821661694762481092699688293, −6.46972429020839511798112126250, −6.20371412498216636535884655763, −6.06424743798278492719366442306, −5.74120433421869074987341278352, −5.31737732229422706927559206562, −4.75066671987062043245986007123, −4.18394956186334251089442632284, −4.16267759114975335419251961844, −3.38708661925172239682050548395, −3.34228521515358622384660519204, −2.96071605181874240499387731422, −2.53635105582756006745090762891, −1.96087733388379321563108560913, −1.55127218323864400537635803381, −1.31086234224679349344492469723, −0.78447760048948505192138535218, 0.78447760048948505192138535218, 1.31086234224679349344492469723, 1.55127218323864400537635803381, 1.96087733388379321563108560913, 2.53635105582756006745090762891, 2.96071605181874240499387731422, 3.34228521515358622384660519204, 3.38708661925172239682050548395, 4.16267759114975335419251961844, 4.18394956186334251089442632284, 4.75066671987062043245986007123, 5.31737732229422706927559206562, 5.74120433421869074987341278352, 6.06424743798278492719366442306, 6.20371412498216636535884655763, 6.46972429020839511798112126250, 7.09821661694762481092699688293, 7.36320937280697382074233550496, 7.71638990036421097168800748589, 8.188321110831117079718253589436

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