↑

Properties

Label 2-520-104.101-c1-0-0
Degree $2$
Conductor $520$
Sign $-0.779 + 0.626i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.795 + 1.16i)2-s + (0.127 − 0.0734i)3-s + (−0.735 + 1.85i)4-s − 5-s + (0.186 + 0.0903i)6-s + (−2.93 − 1.69i)7-s + (−2.76 + 0.617i)8-s + (−1.48 + 2.57i)9-s + (−0.795 − 1.16i)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 + (−2.91 − 2.73i)16-s + (−0.605 + 1.04i)17-s + ⋯
L(s)  = 1  + (0.562 + 0.827i)2-s + (0.0734 − 0.0423i)3-s + (−0.367 + 0.929i)4-s − 0.447·5-s + (0.0763 + 0.0368i)6-s + (−1.11 − 0.640i)7-s + (−0.975 + 0.218i)8-s + (−0.496 + 0.859i)9-s + (−0.251 − 0.369i)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 + (−0.729 − 0.684i)16-s + (−0.146 + 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.779 + 0.626i$
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: \((2,\ 520,\ (\ :1/2),\ -0.779 + 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113683 - 0.323077i\)
\(L(\frac12)\) \(\approx\) \(0.113683 - 0.323077i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.795 - 1.16i)T \)
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 \( 1 + (-7.23 - 4.17i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.35038883569841880655603802462, −10.75977925351272524297862665022, −9.322276908440944342374580739392, −8.633911535152948187485615211707, −7.49579503552679011413912213379, −7.04873706943773583219498406435, −5.79599986229658665628452843055, −5.02162177190491959203231515446, −3.65967961207626334286004197771, −2.91928913734327675068753482693, 0.15419352888156367909545626182, 2.46529496058811849080803211400, 3.20593381431538543129662559165, 4.42998146384279956484382681257, 5.50843224167495216804561714749, 6.40683964567741272061684932364, 7.58787497796510467453780189595, 8.890172877460191981468698702382, 9.732647038750192988648880485268, 10.20247909741573388658519664419

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