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Properties

Label 2-520-520.259-c0-0-2
Degree $2$
Conductor $520$
Sign $0.866 - 0.5i$
Analytic cond. $0.259513$
Root an. cond. $0.509424$
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  − i·2-s + 1.73i·3-s − 4-s + (0.866 − 0.5i)5-s + 1.73·6-s + i·7-s + i·8-s − 1.99·9-s + (−0.5 − 0.866i)10-s − 1.73i·12-s − i·13-s + 14-s + (0.866 + 1.49i)15-s + 16-s + 1.73i·17-s + 1.99i·18-s + ⋯
L(s)  = 1  − i·2-s + 1.73i·3-s − 4-s + (0.866 − 0.5i)5-s + 1.73·6-s + i·7-s + i·8-s − 1.99·9-s + (−0.5 − 0.866i)10-s − 1.73i·12-s − i·13-s + 14-s + (0.866 + 1.49i)15-s + 16-s + 1.73i·17-s + 1.99i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(0.259513\)
Root analytic conductor: \(0.509424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :0),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8798059681\)
\(L(\frac12)\) \(\approx\) \(0.8798059681\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + iT \)
good3 \( 1 - 1.73iT - T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 - 1.73iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
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

−10.69294154625479672328243344568, −10.40933385797328629201130115430, −9.579966444436927184331551592709, −8.816639831550274019315892485694, −8.371524753618588576994694544326, −5.73060836447456813568216832069, −5.49941688629661272568511471686, −4.36372784940494392706454690363, −3.37727247768567768493324725860, −2.17606887471412957133214750858, 1.29435021461707404850600284177, 2.88990503863958712694383797580, 4.67277967118332416632981190486, 5.92072154619868867669469346186, 6.72941149310686717067497387871, 7.14313542959144586051428182778, 7.88635760984614089047252886100, 9.071371019330262090615177276642, 9.854165336521324137789607046417, 11.14102731505417780220091878139

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