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Properties

Label 2-540-540.259-c0-0-1
Degree $2$
Conductor $540$
Sign $0.448 + 0.893i$
Analytic cond. $0.269495$
Root an. cond. $0.519129$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (−0.5 − 0.866i)6-s + (0.326 − 1.85i)7-s + (0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)10-s + (−0.173 + 0.984i)12-s + (−1.43 + 1.20i)14-s + (−0.766 − 0.642i)15-s + (−0.939 + 0.342i)16-s + (−0.173 − 0.984i)18-s + (0.173 − 0.984i)20-s + (0.939 − 1.62i)21-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (−0.5 − 0.866i)6-s + (0.326 − 1.85i)7-s + (0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)10-s + (−0.173 + 0.984i)12-s + (−1.43 + 1.20i)14-s + (−0.766 − 0.642i)15-s + (−0.939 + 0.342i)16-s + (−0.173 − 0.984i)18-s + (0.173 − 0.984i)20-s + (0.939 − 1.62i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.448 + 0.893i$
Analytic conductor: \(0.269495\)
Root analytic conductor: \(0.519129\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :0),\ 0.448 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7596755161\)
\(L(\frac12)\) \(\approx\) \(0.7596755161\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
good7 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
11 \( 1 + (-0.766 + 0.642i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.766 + 0.642i)T^{2} \)
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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

−10.69140019410783199192504893458, −10.11071001601055047861843330588, −9.086872855572790233978400892703, −8.242878860874622578620261023175, −7.57457091391371030696111789793, −7.02040856714404225002644189553, −4.50733394717703212807392311973, −4.05215138524857859605349425699, −3.03421393337947775670832459592, −1.26188146115896189786722346637, 1.96842003330859847024005705051, 3.08380449658343352163709298294, 4.72098932651331337389292228931, 5.95029668030660319716713142076, 6.90095455554735247302232724113, 7.908814786881621315178143746158, 8.406463476776638663088890272125, 9.101351502152284373755888406810, 9.944433777726422813448063538927, 11.23392950384581706170064602955

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