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Properties

Label 2-57330-1.1-c1-0-136
Degree $2$
Conductor $57330$
Sign $1$
Analytic cond. $457.782$
Root an. cond. $21.3958$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s − 4·11-s − 13-s + 16-s − 6·17-s − 4·19-s + 20-s + 4·22-s − 8·23-s + 25-s + 26-s − 6·29-s + 8·31-s − 32-s + 6·34-s − 10·37-s + 4·38-s − 40-s − 6·41-s + 4·43-s − 4·44-s + 8·46-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.20·11-s − 0.277·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.196·26-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 1.64·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s + 1.17·46-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(457.782\)
Root analytic conductor: \(21.3958\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 57330,\ (\ :1/2),\ 1)\)

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$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.95377388957285, −14.46086765055763, −13.68130642428567, −13.37545121706582, −12.94719180071655, −12.22760448838772, −11.78755757714881, −11.23526552264366, −10.52801967234310, −10.20686404895099, −10.00601470500917, −9.018565917445404, −8.722144747919592, −8.262100224017778, −7.594082919174439, −7.086437814585767, −6.532038551745766, −5.878790513299371, −5.499820797475872, −4.619279134046817, −4.194422637115469, −3.280903111827143, −2.415686157332019, −2.198485275584797, −1.437082926842500, 0, 0, 1.437082926842500, 2.198485275584797, 2.415686157332019, 3.280903111827143, 4.194422637115469, 4.619279134046817, 5.499820797475872, 5.878790513299371, 6.532038551745766, 7.086437814585767, 7.594082919174439, 8.262100224017778, 8.722144747919592, 9.018565917445404, 10.00601470500917, 10.20686404895099, 10.52801967234310, 11.23526552264366, 11.78755757714881, 12.22760448838772, 12.94719180071655, 13.37545121706582, 13.68130642428567, 14.46086765055763, 14.95377388957285

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