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Properties

Label 2-520-520.19-c1-0-61
Degree $2$
Conductor $520$
Sign $0.360 + 0.932i$
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  + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + (−1.58 + 1.58i)5-s + (1.21 − 4.52i)7-s + (−2 − 1.99i)8-s + (−1.5 − 2.59i)9-s + (−2.73 − 1.58i)10-s + (−5.40 + 1.44i)11-s + (3.31 − 1.42i)13-s + 6.62·14-s + (1.99 − 3.46i)16-s + (3 − 3i)18-s + (−7.17 − 1.92i)19-s + (1.15 − 4.31i)20-s + (−3.95 − 6.84i)22-s + (2.45 + 1.41i)23-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + (−0.707 + 0.707i)5-s + (0.458 − 1.71i)7-s + (−0.707 − 0.707i)8-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.499i)10-s + (−1.62 + 0.436i)11-s + (0.918 − 0.396i)13-s + 1.77·14-s + (0.499 − 0.866i)16-s + (0.707 − 0.707i)18-s + (−1.64 − 0.441i)19-s + (0.258 − 0.965i)20-s + (−0.842 − 1.46i)22-s + (0.512 + 0.295i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.360 + 0.932i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.360 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446796 - 0.306406i\)
\(L(\frac12)\) \(\approx\) \(0.446796 - 0.306406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
5 \( 1 + (1.58 - 1.58i)T \)
13 \( 1 + (-3.31 + 1.42i)T \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.21 + 4.52i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (5.40 - 1.44i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.17 + 1.92i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.45 - 1.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 31iT^{2} \)
37 \( 1 + (1.72 - 0.461i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.68 - 10.0i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.21 + 9.21i)T + 47iT^{2} \)
53 \( 1 + 1.99T + 53T^{2} \)
59 \( 1 + (-3.71 + 13.8i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (9.89 - 2.65i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-84.0 - 48.5i)T^{2} \)
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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

−10.72750876863354196541852131523, −9.919467870384182677399247920492, −8.384723742329837368251544296578, −7.961609597627802908061235584827, −7.01400069677159219326220320171, −6.39105795302584105522323126547, −4.97411743734100834856815234943, −4.03865994827484368550851699197, −3.16050946283080642443000178636, −0.28403694101476584671396543041, 1.97589581859079780770248622578, 2.91063914831399036168584318661, 4.40761285848654430619637514484, 5.29698626698009235686465550595, 5.87713292284355460542733347199, 8.056136955746209620383119708210, 8.493442905764449185859188605165, 9.041395030762206476216769042527, 10.58430461339287721663687425161, 11.07560650553122303396170753504

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