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Properties

Label 2-520-520.179-c0-0-0
Degree $2$
Conductor $520$
Sign $-0.488 - 0.872i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−1.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (1.5 + 0.866i)19-s + (0.866 + 0.499i)20-s + (0.866 − 1.5i)22-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−1.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (1.5 + 0.866i)19-s + (0.866 + 0.499i)20-s + (0.866 − 1.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.488 - 0.872i$
Analytic conductor: \(0.259513\)
Root analytic conductor: \(0.509424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :0),\ -0.488 - 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4566775933\)
\(L(\frac12)\) \(\approx\) \(0.4566775933\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 - iT \)
13 \( 1 + (-0.866 - 0.5i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 - 1.73T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−11.01831507430911941842172410708, −10.17450334272026736478646852483, −9.985842344143687729069548426577, −8.595167834695629082271921204311, −7.59888857193649313895680098096, −7.14685092221581406992389449569, −6.05997673080311100633987975301, −5.15957785718014359543120775760, −3.34762850506061409269039838184, −2.15120093263783649650930372931, 0.72868352917303411459254917018, 2.78026838152998670291913947864, 3.53704010300855257745047014038, 5.34269000573217869409882753255, 6.13500900540880393132653599213, 7.49676643731216824792355015602, 8.461025563780525002572026677015, 9.004313165278166771512836353829, 9.738186019940106128938094282264, 10.75085712535646964254439033972

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