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Properties

Label 2-513-171.49-c1-0-15
Degree $2$
Conductor $513$
Sign $-0.453 + 0.891i$
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  + 1.23·2-s − 0.478·4-s + (−0.275 − 0.477i)5-s + (−1.62 − 2.80i)7-s − 3.05·8-s + (−0.340 − 0.589i)10-s + (−2.68 − 4.65i)11-s + 3.52·13-s + (−2.00 − 3.46i)14-s − 2.81·16-s + (−2.60 + 4.50i)17-s + (0.164 − 4.35i)19-s + (0.131 + 0.228i)20-s + (−3.31 − 5.74i)22-s − 2.98·23-s + ⋯
L(s)  = 1  + 0.872·2-s − 0.239·4-s + (−0.123 − 0.213i)5-s + (−0.612 − 1.06i)7-s − 1.08·8-s + (−0.107 − 0.186i)10-s + (−0.810 − 1.40i)11-s + 0.977·13-s + (−0.534 − 0.925i)14-s − 0.703·16-s + (−0.630 + 1.09i)17-s + (0.0376 − 0.999i)19-s + (0.0294 + 0.0510i)20-s + (−0.706 − 1.22i)22-s − 0.622·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.453 + 0.891i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ -0.453 + 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.628213 - 1.02476i\)
\(L(\frac12)\) \(\approx\) \(0.628213 - 1.02476i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.164 + 4.35i)T \)
good2 \( 1 - 1.23T + 2T^{2} \)
5 \( 1 + (0.275 + 0.477i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.62 + 2.80i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.68 + 4.65i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.52T + 13T^{2} \)
17 \( 1 + (2.60 - 4.50i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + 2.98T + 23T^{2} \)
29 \( 1 + (-2.74 + 4.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.54 - 4.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.20T + 37T^{2} \)
41 \( 1 + (-0.855 - 1.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 3.59T + 43T^{2} \)
47 \( 1 + (5.31 - 9.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.562 + 0.973i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.88 + 6.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.68 + 9.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 2.37T + 67T^{2} \)
71 \( 1 + (-0.507 + 0.879i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.98 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.13T + 79T^{2} \)
83 \( 1 + (-1.14 - 1.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.12 + 1.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.18T + 97T^{2} \)
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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

−10.80505012868857952771087527917, −9.807659099141465389011766015075, −8.649546138743197402834858404844, −8.091576247976130341462505654366, −6.51014367856309756685685968437, −6.00060954557410459024593906351, −4.70538229903782576970174927080, −3.86334346934596629186318008836, −2.97886387552794123377339022082, −0.53068899968814273337140194172, 2.36651451714521523923051166795, 3.42001130599133256837886904014, 4.58721647823269735039363104592, 5.49249426152670279344935059542, 6.30956797358469083527278395053, 7.43644186818854535972397861350, 8.652190663254864857942893759338, 9.397234421909703685903684941588, 10.21775637761256346626051854473, 11.50555410121039879802640398500

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