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Properties

Label 2-520-520.419-c0-0-1
Degree $2$
Conductor $520$
Sign $0.711 + 0.702i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 5-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + 0.999·18-s + (0.5 − 0.866i)19-s + (−0.499 + 0.866i)20-s + (0.499 − 0.866i)22-s + (−1 − 1.73i)23-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 5-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + 0.999·18-s + (0.5 − 0.866i)19-s + (−0.499 + 0.866i)20-s + (0.499 − 0.866i)22-s + (−1 − 1.73i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(0.259513\)
Root analytic conductor: \(0.509424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :0),\ 0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7966672755\)
\(L(\frac12)\) \(\approx\) \(0.7966672755\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.77067965731747942369156108531, −10.20539495146583770430467638039, −9.399579583385886081768881141901, −8.538466092957520657842002535368, −7.49551867066761914812030388336, −6.62471132732639840681293628213, −4.95376478752803301222223787892, −4.33336487180665466275207594290, −2.59546660216884216447239519965, −1.68817120029887421572417708463, 1.59941800134985132731665511997, 3.32082361733897575801104037403, 5.14900154662312673340487555505, 5.82308008856951417349663755863, 6.35362586849803038207357510147, 7.79602510967731700810631773484, 8.491636354940736302731446810811, 9.482937725342244738769472244672, 9.812403874490014671307762153877, 11.11431358828434082847527430276

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