Properties

Label 4-39e4-1.1-c1e2-0-0
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $147.507$
Root an. cond. $3.48500$
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  + 2-s + 4-s − 3·5-s + 3·7-s + 3·8-s − 3·10-s − 4·11-s + 3·14-s + 16-s − 17-s − 6·19-s − 3·20-s − 4·22-s − 4·23-s + 25-s + 3·28-s − 29-s − 31-s − 32-s − 34-s − 9·35-s − 11·37-s − 6·38-s − 9·40-s + 41-s + 5·43-s − 4·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.34·5-s + 1.13·7-s + 1.06·8-s − 0.948·10-s − 1.20·11-s + 0.801·14-s + 1/4·16-s − 0.242·17-s − 1.37·19-s − 0.670·20-s − 0.852·22-s − 0.834·23-s + 1/5·25-s + 0.566·28-s − 0.185·29-s − 0.179·31-s − 0.176·32-s − 0.171·34-s − 1.52·35-s − 1.80·37-s − 0.973·38-s − 1.42·40-s + 0.156·41-s + 0.762·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(147.507\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2313441,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.602008771\)
\(L(\frac12)\) \(\approx\) \(1.602008771\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good2$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - T + 78 T^{2} - p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T + 88 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 11 T + 98 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 16 T + 169 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 5 T + 136 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 12 T + 165 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 15 T + 210 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 10 T + 174 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 18 T + 242 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 13 T + 232 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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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

−9.657833910237604544020920023976, −9.263489553812535001502362649945, −8.608679826875966097356259203068, −8.300635451467443019539945326496, −7.921174208219334450299661492516, −7.73931305784082476419620846140, −7.48121404536555195585443578199, −6.71767869519000070985174829874, −6.58540375912215168389480304264, −5.96063859230283993864858538008, −5.23574649215559206737936130433, −5.04062066361909862678800800901, −4.67577561131754230408858777349, −4.23872030791686903669152338881, −3.65157862846727552304585172851, −3.56797210881569497076903541704, −2.52774417137176119618141532376, −2.06902170765143710046164267312, −1.70604796692066655437177248167, −0.42231529706299085034984012268, 0.42231529706299085034984012268, 1.70604796692066655437177248167, 2.06902170765143710046164267312, 2.52774417137176119618141532376, 3.56797210881569497076903541704, 3.65157862846727552304585172851, 4.23872030791686903669152338881, 4.67577561131754230408858777349, 5.04062066361909862678800800901, 5.23574649215559206737936130433, 5.96063859230283993864858538008, 6.58540375912215168389480304264, 6.71767869519000070985174829874, 7.48121404536555195585443578199, 7.73931305784082476419620846140, 7.921174208219334450299661492516, 8.300635451467443019539945326496, 8.608679826875966097356259203068, 9.263489553812535001502362649945, 9.657833910237604544020920023976

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