Properties

Label 4-39e4-1.1-c1e2-0-6
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 12·16-s + 10·25-s − 26·43-s − 13·49-s + 26·61-s + 32·64-s + 26·79-s + 40·100-s − 26·103-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·172-s + 173-s + 179-s + 181-s + 191-s + 193-s − 52·196-s + ⋯
L(s)  = 1  + 2·4-s + 3·16-s + 2·25-s − 3.96·43-s − 1.85·49-s + 3.32·61-s + 4·64-s + 2.92·79-s + 4·100-s − 2.56·103-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 7.92·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 3.71·196-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

\(L(1)\) \(\approx\) \(4.417643070\)
\(L(\frac12)\) \(\approx\) \(4.417643070\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−9.606834570620730067007222022531, −9.562659418411670609560770303852, −8.634963774147364067892326096121, −8.301313944502768042012743912665, −8.169273316086556433005792833824, −7.63591482532445221823106186314, −6.95134974112846520078597538902, −6.89467014712338490590698282284, −6.50618795599704138466893898084, −6.32330957485005699752075428604, −5.50415515788240422222198479319, −5.14498838884368396271322134815, −4.94242111528745808001406511268, −4.01797065380307853057820167715, −3.36476624665477288762519080122, −3.20749209732832894567563078283, −2.63157065887121620480903494219, −1.99078763810558088228352808591, −1.60843659544344408192544238218, −0.823052652157452310745906884638, 0.823052652157452310745906884638, 1.60843659544344408192544238218, 1.99078763810558088228352808591, 2.63157065887121620480903494219, 3.20749209732832894567563078283, 3.36476624665477288762519080122, 4.01797065380307853057820167715, 4.94242111528745808001406511268, 5.14498838884368396271322134815, 5.50415515788240422222198479319, 6.32330957485005699752075428604, 6.50618795599704138466893898084, 6.89467014712338490590698282284, 6.95134974112846520078597538902, 7.63591482532445221823106186314, 8.169273316086556433005792833824, 8.301313944502768042012743912665, 8.634963774147364067892326096121, 9.562659418411670609560770303852, 9.606834570620730067007222022531

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