Properties

Label 4-7623e2-1.1-c1e2-0-31
Degree $4$
Conductor $58110129$
Sign $1$
Analytic cond. $3705.15$
Root an. cond. $7.80192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·4-s + 3·5-s + 2·7-s + 3·8-s − 3·10-s − 2·13-s − 2·14-s + 16-s + 4·17-s − 6·20-s + 2·23-s − 2·25-s + 2·26-s − 4·28-s − 16·31-s − 2·32-s − 4·34-s + 6·35-s − 4·37-s + 9·40-s + 41-s − 2·43-s − 2·46-s − 47-s + 3·49-s + 2·50-s + ⋯
L(s)  = 1  − 0.707·2-s − 4-s + 1.34·5-s + 0.755·7-s + 1.06·8-s − 0.948·10-s − 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s − 1.34·20-s + 0.417·23-s − 2/5·25-s + 0.392·26-s − 0.755·28-s − 2.87·31-s − 0.353·32-s − 0.685·34-s + 1.01·35-s − 0.657·37-s + 1.42·40-s + 0.156·41-s − 0.304·43-s − 0.294·46-s − 0.145·47-s + 3/7·49-s + 0.282·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(58110129\)    =    \(3^{4} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(3705.15\)
Root analytic conductor: \(7.80192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 58110129,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 16 T + 121 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - T + 51 T^{2} - p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 13 T + 147 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 15 T + 163 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 181 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 6 T + 131 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 15 T + 203 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 15 T + 233 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \)
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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

−7.74607574496326732163732655815, −7.69268633393631913841493159831, −6.95629815061482692299567982049, −6.80293717043010550500974724616, −6.31253448537302011263069804746, −5.79973212324884226975447910529, −5.44421402194162496616611601776, −5.40949830285311885125023703007, −4.87855170042502645689574655224, −4.77250187209593489514959980178, −3.96634457488034567424463367243, −3.87219579206636883680054317876, −3.40356513863813462369537628487, −2.77788825050618481757192340986, −2.30442816032702091542250214980, −1.88414502760823338474179411097, −1.33183085835872365449060862032, −1.26301442173657144658398263786, 0, 0, 1.26301442173657144658398263786, 1.33183085835872365449060862032, 1.88414502760823338474179411097, 2.30442816032702091542250214980, 2.77788825050618481757192340986, 3.40356513863813462369537628487, 3.87219579206636883680054317876, 3.96634457488034567424463367243, 4.77250187209593489514959980178, 4.87855170042502645689574655224, 5.40949830285311885125023703007, 5.44421402194162496616611601776, 5.79973212324884226975447910529, 6.31253448537302011263069804746, 6.80293717043010550500974724616, 6.95629815061482692299567982049, 7.69268633393631913841493159831, 7.74607574496326732163732655815

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