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Properties

Label 2-513-19.5-c1-0-4
Degree $2$
Conductor $513$
Sign $-0.600 - 0.799i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
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  + (−2.37 + 0.866i)2-s + (3.37 − 2.83i)4-s + (1.03 + 0.866i)5-s + (2.43 + 4.22i)7-s + (−3.05 + 5.28i)8-s + (−3.20 − 1.16i)10-s + (−2.09 + 3.62i)11-s + (−0.460 − 2.61i)13-s + (−9.46 − 7.94i)14-s + (1.15 − 6.53i)16-s + (6.12 − 2.22i)17-s + (−2.52 + 3.55i)19-s + 5.94·20-s + (1.84 − 10.4i)22-s + (−1.93 + 1.62i)23-s + ⋯
L(s)  = 1  + (−1.68 + 0.612i)2-s + (1.68 − 1.41i)4-s + (0.461 + 0.387i)5-s + (0.922 + 1.59i)7-s + (−1.07 + 1.86i)8-s + (−1.01 − 0.368i)10-s + (−0.630 + 1.09i)11-s + (−0.127 − 0.724i)13-s + (−2.52 − 2.12i)14-s + (0.288 − 1.63i)16-s + (1.48 − 0.540i)17-s + (−0.578 + 0.815i)19-s + 1.32·20-s + (0.392 − 2.22i)22-s + (−0.404 + 0.339i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.600 - 0.799i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ -0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.311179 + 0.622751i\)
\(L(\frac12)\) \(\approx\) \(0.311179 + 0.622751i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (2.52 - 3.55i)T \)
good2 \( 1 + (2.37 - 0.866i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-1.03 - 0.866i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-2.43 - 4.22i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.09 - 3.62i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.460 + 2.61i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-6.12 + 2.22i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (1.93 - 1.62i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-2.76 - 1.00i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.16 - 5.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.24T + 37T^{2} \)
41 \( 1 + (-1.27 + 7.23i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (2.51 + 2.11i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (1.41 + 0.516i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (3.05 - 2.56i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (3.35 - 1.22i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.28 + 1.91i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.37 - 0.502i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-0.343 - 0.288i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.54 - 8.78i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.74 - 9.87i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.29 - 3.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.24 - 18.3i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-13.6 + 4.96i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.67349530085324273803657379481, −10.13338030187837626495367370336, −9.425124965993172477549719112999, −8.325703826383555572674185037401, −7.983031999808624918919690840002, −6.92942520969392794803136314760, −5.78171723314774197194461276699, −5.20251224361130414879956048341, −2.62729525206875208247667096859, −1.70070076871972315885116248648, 0.74065978201454403718245220722, 1.78127198673469775549520262958, 3.33849507085554099759583968381, 4.73209309563653866331118300750, 6.29172844867289260410689947904, 7.50011776815923060653109890687, 8.027406216618545492945116625818, 8.801618358984267966479054195663, 9.894413293323953181178538019605, 10.40754303505679723682729283548

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