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Properties

Label 2-520-520.259-c0-0-1
Degree $2$
Conductor $520$
Sign $-i$
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 − 4-s − i·5-s + 2i·7-s − i·8-s + 9-s + 10-s + i·13-s − 2·14-s + 16-s + i·18-s + i·20-s − 25-s − 26-s − 2i·28-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·5-s + 2i·7-s − i·8-s + 9-s + 10-s + i·13-s − 2·14-s + 16-s + i·18-s + i·20-s − 25-s − 26-s − 2i·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-i$
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: \((2,\ 520,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8389854345\)
\(L(\frac12)\) \(\approx\) \(0.8389854345\)
\(L(1)\) 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 \( 1 + iT \)
13 \( 1 - iT \)
good3 \( 1 - T^{2} \)
7 \( 1 - 2iT - T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 2iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
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

−11.61144210683494560605853591150, −9.967725228766571548974597952849, −9.097366550282792403995258915226, −8.835719031342380150876928511560, −7.79071595391254566375437983893, −6.66895334375232432268466879257, −5.68442639648391879796804776675, −5.00011075568784460688621057559, −3.96837978668132947015585791747, −1.94812376386566031223050970441, 1.27353065133542307062269247194, 3.03051560711386579805053572207, 3.89645195850844558732101743240, 4.76106978250826482717527921883, 6.40766536516656307691879715818, 7.42000898306740115833714573274, 8.011246232167615518093644075374, 9.680536922925315456174638285980, 10.22252474658713471480439665398, 10.68997986242741319590014741388

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