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Properties

Label 2-57330-1.1-c1-0-38
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 $0$

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 + 8·17-s + 6·19-s − 20-s + 4·22-s − 6·23-s + 25-s − 26-s + 4·29-s − 32-s − 8·34-s − 2·37-s − 6·38-s + 40-s − 2·41-s − 4·43-s − 4·44-s + 6·46-s − 50-s + 52-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 + 1.94·17-s + 1.37·19-s − 0.223·20-s + 0.852·22-s − 1.25·23-s + 1/5·25-s − 0.196·26-s + 0.742·29-s − 0.176·32-s − 1.37·34-s − 0.328·37-s − 0.973·38-s + 0.158·40-s − 0.312·41-s − 0.609·43-s − 0.603·44-s + 0.884·46-s − 0.141·50-s + 0.138·52-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: \(0\)
Selberg data: \((2,\ 57330,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.611877284\)
\(L(\frac12)\) \(\approx\) \(1.611877284\)
\(L(\frac{3}{2})\) 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 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−14.42413975579625, −13.82167564341635, −13.48502014710091, −12.61808004233884, −12.24427581841400, −11.81148997477204, −11.33806122856358, −10.63986056311741, −10.11684049357727, −9.900181372110497, −9.321392410899556, −8.413748946851438, −8.071650335922281, −7.821970441580602, −7.141385815505200, −6.646256997673342, −5.734480491997853, −5.402710073183489, −4.900204202022254, −3.745145027938324, −3.497320790983493, −2.723401223170884, −2.076176029815504, −1.094045164260567, −0.5733147551590123, 0.5733147551590123, 1.094045164260567, 2.076176029815504, 2.723401223170884, 3.497320790983493, 3.745145027938324, 4.900204202022254, 5.402710073183489, 5.734480491997853, 6.646256997673342, 7.141385815505200, 7.821970441580602, 8.071650335922281, 8.413748946851438, 9.321392410899556, 9.900181372110497, 10.11684049357727, 10.63986056311741, 11.33806122856358, 11.81148997477204, 12.24427581841400, 12.61808004233884, 13.48502014710091, 13.82167564341635, 14.42413975579625

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