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Properties

Label 2-520-65.63-c1-0-9
Degree $2$
Conductor $520$
Sign $0.999 - 0.0313i$
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  + (−0.593 + 2.21i)3-s + (0.325 − 2.21i)5-s + (−3.90 − 2.25i)7-s + (−1.94 − 1.12i)9-s + (6.02 + 1.61i)11-s + (2.77 − 2.30i)13-s + (4.70 + 2.03i)15-s + (4.04 − 1.08i)17-s + (0.0419 + 0.156i)19-s + (7.31 − 7.31i)21-s + (1.68 + 0.451i)23-s + (−4.78 − 1.43i)25-s + (−1.21 + 1.21i)27-s + (2.00 − 1.15i)29-s + (3.80 + 3.80i)31-s + ⋯
L(s)  = 1  + (−0.342 + 1.27i)3-s + (0.145 − 0.989i)5-s + (−1.47 − 0.853i)7-s + (−0.649 − 0.375i)9-s + (1.81 + 0.486i)11-s + (0.769 − 0.638i)13-s + (1.21 + 0.524i)15-s + (0.980 − 0.262i)17-s + (0.00962 + 0.0359i)19-s + (1.59 − 1.59i)21-s + (0.351 + 0.0941i)23-s + (−0.957 − 0.287i)25-s + (−0.233 + 0.233i)27-s + (0.371 − 0.214i)29-s + (0.683 + 0.683i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.999 - 0.0313i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.999 - 0.0313i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26532 + 0.0198102i\)
\(L(\frac12)\) \(\approx\) \(1.26532 + 0.0198102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.325 + 2.21i)T \)
13 \( 1 + (-2.77 + 2.30i)T \)
good3 \( 1 + (0.593 - 2.21i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (3.90 + 2.25i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-6.02 - 1.61i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-4.04 + 1.08i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.0419 - 0.156i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.68 - 0.451i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.00 + 1.15i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.80 - 3.80i)T + 31iT^{2} \)
37 \( 1 + (-2.59 + 1.49i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.462 + 1.72i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.90 + 10.8i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 4.43iT - 47T^{2} \)
53 \( 1 + (4.24 + 4.24i)T + 53iT^{2} \)
59 \( 1 + (-6.94 + 1.86i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.31 - 4.01i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.15 - 5.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.00888 + 0.00238i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + 9.58iT - 79T^{2} \)
83 \( 1 - 9.50iT - 83T^{2} \)
89 \( 1 + (0.433 - 1.61i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.14 - 7.18i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.57528645000476486436501811555, −9.854429314400191354635140283460, −9.449602585276796051135723869315, −8.556980833621660847318690474398, −7.06890857569200926362690696014, −6.13989270462596909063968012493, −5.11785636887723537584596572477, −3.97407713187104144802381494511, −3.55925953202559852807277285741, −0.968043681430727545119949351099, 1.31684443381597718802804555592, 2.81658010654705558612515321498, 3.77774042394490831979749954107, 6.00657323490093694559951816937, 6.35185830280097429720999776729, 6.79584129200127036913814004265, 8.040408766106529100093235575337, 9.224451566941158235359630940689, 9.808533195845397816868079117606, 11.20812079013735279736211436494

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