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Properties

Label 2-513-171.49-c1-0-10
Degree $2$
Conductor $513$
Sign $0.0354 + 0.999i$
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  − 2.46·2-s + 4.09·4-s + (0.957 + 1.65i)5-s + (−0.324 − 0.562i)7-s − 5.16·8-s + (−2.36 − 4.09i)10-s + (−2.93 − 5.07i)11-s − 0.655·13-s + (0.801 + 1.38i)14-s + 4.55·16-s + (1.93 − 3.35i)17-s + (−4.28 − 0.802i)19-s + (3.91 + 6.78i)20-s + (7.23 + 12.5i)22-s + 1.92·23-s + ⋯
L(s)  = 1  − 1.74·2-s + 2.04·4-s + (0.428 + 0.741i)5-s + (−0.122 − 0.212i)7-s − 1.82·8-s + (−0.747 − 1.29i)10-s + (−0.884 − 1.53i)11-s − 0.181·13-s + (0.214 + 0.370i)14-s + 1.13·16-s + (0.469 − 0.813i)17-s + (−0.982 − 0.184i)19-s + (0.876 + 1.51i)20-s + (1.54 + 2.67i)22-s + 0.401·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0354 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0354 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.0354 + 0.999i$
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: \((2,\ 513,\ (\ :1/2),\ 0.0354 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.320098 - 0.308936i\)
\(L(\frac12)\) \(\approx\) \(0.320098 - 0.308936i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.28 + 0.802i)T \)
good2 \( 1 + 2.46T + 2T^{2} \)
5 \( 1 + (-0.957 - 1.65i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.324 + 0.562i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.93 + 5.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.655T + 13T^{2} \)
17 \( 1 + (-1.93 + 3.35i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 - 1.92T + 23T^{2} \)
29 \( 1 + (3.26 - 5.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.54 + 2.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.23T + 37T^{2} \)
41 \( 1 + (3.48 + 6.03i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 8.93T + 43T^{2} \)
47 \( 1 + (-5.77 + 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.35 - 11.0i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.16 + 12.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.17 + 8.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 0.763T + 67T^{2} \)
71 \( 1 + (-0.299 + 0.519i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.75 + 3.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.26T + 79T^{2} \)
83 \( 1 + (-3.29 - 5.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.41 + 4.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.38T + 97T^{2} \)
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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

−10.63686175749643666989663847587, −9.837455238785164890945631753616, −8.859225667939148754743415650899, −8.199878800501428392855127849423, −7.20661860959244454523211414365, −6.50132005280131604179903757073, −5.38811165971390418864252522413, −3.25442752153885145630332160316, −2.26409598968594238671580131266, −0.46050349682364705284372042454, 1.48336087917558305885832080784, 2.49545295699788172452794663000, 4.54621381633402133734224198091, 5.78445435975146999028817818191, 6.92190289192246315718132065118, 7.80016729065729306026101195805, 8.522865226944195017625414149708, 9.395018422944915199243797320743, 10.03653891435183925917469771131, 10.61821647342023765512694300373

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