Properties

Label 4-735e2-1.1-c1e2-0-1
Degree $4$
Conductor $540225$
Sign $1$
Analytic cond. $34.4452$
Root an. cond. $2.42260$
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  + 3·4-s − 2·5-s − 9-s − 12·11-s + 5·16-s − 12·19-s − 6·20-s − 25-s + 4·29-s + 20·31-s − 3·36-s − 4·41-s − 36·44-s + 2·45-s + 24·55-s − 16·59-s + 4·61-s + 3·64-s + 20·71-s − 36·76-s − 8·79-s − 10·80-s + 81-s + 12·89-s + 24·95-s + 12·99-s − 3·100-s + ⋯
L(s)  = 1  + 3/2·4-s − 0.894·5-s − 1/3·9-s − 3.61·11-s + 5/4·16-s − 2.75·19-s − 1.34·20-s − 1/5·25-s + 0.742·29-s + 3.59·31-s − 1/2·36-s − 0.624·41-s − 5.42·44-s + 0.298·45-s + 3.23·55-s − 2.08·59-s + 0.512·61-s + 3/8·64-s + 2.37·71-s − 4.12·76-s − 0.900·79-s − 1.11·80-s + 1/9·81-s + 1.27·89-s + 2.46·95-s + 1.20·99-s − 0.299·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(540225\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(34.4452\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{735} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 540225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.013321091\)
\(L(\frac12)\) \(\approx\) \(1.013321091\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−10.65432352920307798340698525594, −10.26264449642343933639969614141, −10.24954674872127833581824633607, −9.390474647614821434448921376422, −8.390750672063355757797823404983, −8.262007300807147454324838074952, −8.118093345649354850343782386613, −7.71382622968980540644197573214, −7.18107025616131508097129846225, −6.50068627462598728544158889301, −6.39823248084765288028849059082, −5.79385320903557292475169219793, −5.20980453827603346495185123115, −4.50967170295167665734040054949, −4.49572321492708365576392047484, −3.23140721448490455171248779770, −2.92714915453075174170310452548, −2.32101591337753869525714945934, −2.15426545704181043667133261772, −0.46777483556197831922160617880, 0.46777483556197831922160617880, 2.15426545704181043667133261772, 2.32101591337753869525714945934, 2.92714915453075174170310452548, 3.23140721448490455171248779770, 4.49572321492708365576392047484, 4.50967170295167665734040054949, 5.20980453827603346495185123115, 5.79385320903557292475169219793, 6.39823248084765288028849059082, 6.50068627462598728544158889301, 7.18107025616131508097129846225, 7.71382622968980540644197573214, 8.118093345649354850343782386613, 8.262007300807147454324838074952, 8.390750672063355757797823404983, 9.390474647614821434448921376422, 10.24954674872127833581824633607, 10.26264449642343933639969614141, 10.65432352920307798340698525594

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