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Properties

Label 2-520-520.109-c0-0-0
Degree $2$
Conductor $520$
Sign $0.471 - 0.881i$
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  + (−0.707 + 0.707i)2-s + 1.41·3-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + (−1.00 + 1.00i)6-s + (0.707 + 0.707i)8-s + 1.00·9-s − 1.00i·10-s − 1.41i·12-s + (0.707 + 0.707i)13-s + (−1.00 + 1.00i)15-s − 1.00·16-s + (−0.707 + 0.707i)18-s + (0.707 + 0.707i)20-s + (1.00 + 1.00i)24-s − 1.00i·25-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + 1.41·3-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + (−1.00 + 1.00i)6-s + (0.707 + 0.707i)8-s + 1.00·9-s − 1.00i·10-s − 1.41i·12-s + (0.707 + 0.707i)13-s + (−1.00 + 1.00i)15-s − 1.00·16-s + (−0.707 + 0.707i)18-s + (0.707 + 0.707i)20-s + (1.00 + 1.00i)24-s − 1.00i·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(0.259513\)
Root analytic conductor: \(0.509424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :0),\ 0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8826792484\)
\(L(\frac12)\) \(\approx\) \(0.8826792484\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 - 1.41T + T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1 + i)T - iT^{2} \)
37 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
41 \( 1 + (1 - i)T - iT^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
71 \( 1 + (-1 + i)T - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
89 \( 1 + (1 + i)T + iT^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.97878330071379521237376372687, −10.11210742672307641836459612889, −9.157369274406247467272798639635, −8.519766234379053400348560270996, −7.77314616423048239091256931840, −7.04987121624764931831776443476, −6.08297200252352400579838337665, −4.43032339256176632506420511539, −3.34248166009934624772660620541, −2.02078760880861177854780019502, 1.51150197739958982890842719183, 3.03953977040477767033589900902, 3.66379028499651929801667774611, 4.88265770832137033641126062981, 6.82316652502474154847826330132, 8.013694403214366750232369857092, 8.291875519969350522920555649616, 9.001300333654929691889224913798, 9.875591505012018227044014420792, 10.79630250139463090406828126889

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