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Properties

Label 2-540-20.19-c0-0-3
Degree $2$
Conductor $540$
Sign $0.866 + 0.5i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s − 0.999i·8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)16-s + i·17-s − 1.73i·19-s + (0.866 + 0.499i)20-s − 1.73·23-s − 25-s + 1.73i·31-s + (−0.866 − 0.499i)32-s + (0.5 + 0.866i)34-s + (−0.866 − 1.49i)38-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s − 0.999i·8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)16-s + i·17-s − 1.73i·19-s + (0.866 + 0.499i)20-s − 1.73·23-s − 25-s + 1.73i·31-s + (−0.866 − 0.499i)32-s + (0.5 + 0.866i)34-s + (−0.866 − 1.49i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(0.269495\)
Root analytic conductor: \(0.519129\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :0),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.386696112\)
\(L(\frac12)\) \(\approx\) \(1.386696112\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + 1.73T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 - 1.73T + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.00429158635967935766255552748, −10.37083100554285840541450862264, −9.551758409027022399739321536258, −8.227315168474750093171516355143, −6.96215323502674690653887542145, −6.39259010540060493263700033141, −5.30037614698566183435265238598, −4.11522237234904982467074625773, −3.13662040918974311115975306753, −2.01934641242289609555372219838, 2.05919353424284405404061905360, 3.68449568480153973145395404011, 4.51620722251525995151590446071, 5.57961588409205093278924305213, 6.23152925981884095738600555322, 7.68354224365368074276537993379, 8.084639044163893863602298178689, 9.247391330770179373641726798397, 10.17165415983521774831511962616, 11.57048636574330633985988260267

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