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Properties

Label 2-520-104.101-c1-0-13
Degree $2$
Conductor $520$
Sign $0.972 - 0.233i$
Analytic cond. $4.15222$
Root an. cond. $2.03769$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.615 − 1.27i)2-s + (−0.127 + 0.0734i)3-s + (−1.24 + 1.56i)4-s + 5-s + (0.171 + 0.116i)6-s + (−2.93 − 1.69i)7-s + (2.76 + 0.617i)8-s + (−1.48 + 2.57i)9-s + (−0.615 − 1.27i)10-s + (2.72 + 4.72i)11-s + (0.0429 − 0.290i)12-s + (1.96 − 3.01i)13-s + (−0.351 + 4.78i)14-s + (−0.127 + 0.0734i)15-s + (−0.911 − 3.89i)16-s + (−0.605 + 1.04i)17-s + ⋯
L(s)  = 1  + (−0.435 − 0.900i)2-s + (−0.0734 + 0.0423i)3-s + (−0.621 + 0.783i)4-s + 0.447·5-s + (0.0701 + 0.0476i)6-s + (−1.11 − 0.640i)7-s + (0.975 + 0.218i)8-s + (−0.496 + 0.859i)9-s + (−0.194 − 0.402i)10-s + (0.821 + 1.42i)11-s + (0.0124 − 0.0838i)12-s + (0.546 − 0.837i)13-s + (−0.0940 + 1.27i)14-s + (−0.0328 + 0.0189i)15-s + (−0.227 − 0.973i)16-s + (−0.146 + 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.972 - 0.233i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.972 - 0.233i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.919689 + 0.109110i\)
\(L(\frac12)\) \(\approx\) \(0.919689 + 0.109110i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.615 + 1.27i)T \)
5 \( 1 - T \)
13 \( 1 + (-1.96 + 3.01i)T \)
good3 \( 1 + (0.127 - 0.0734i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (2.93 + 1.69i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.72 - 4.72i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.605 - 1.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.73 - 3.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.76 - 6.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.10 + 3.52i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.60iT - 31T^{2} \)
37 \( 1 + (-3.82 - 6.62i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.47 + 2.00i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.45 + 0.841i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.231iT - 47T^{2} \)
53 \( 1 + 5.16iT - 53T^{2} \)
59 \( 1 + (1.57 - 2.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.73 - 4.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.31 + 7.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (13.9 + 8.05i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.1iT - 73T^{2} \)
79 \( 1 - 6.72T + 79T^{2} \)
83 \( 1 - 5.31T + 83T^{2} \)
89 \( 1 + (8.75 - 5.05i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.23 - 4.17i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.61053363384611251462164927713, −10.14391725509547188687931430295, −9.438424660600124487287438869534, −8.428400044393655265662666919493, −7.42128913618075521812661889681, −6.42832018732874668654135931764, −5.05859554439950563673135243455, −3.91078642967024155841616901044, −2.85093360255977855582699811153, −1.43739499758107553542587657083, 0.70482798845407586880106934171, 2.87086380099769589696957779339, 4.23478062363591679304462358788, 5.77316163353839840285157021592, 6.33380313077150501699720047014, 6.78105655766776464751469306996, 8.508426688111045688884127249452, 9.043382240712240118277642028403, 9.442793546171058397160939944775, 10.75229998936710925501384024730

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