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Properties

Label 2-513-171.49-c1-0-7
Degree $2$
Conductor $513$
Sign $0.814 - 0.580i$
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  + 1.69·2-s + 0.870·4-s + (0.0441 + 0.0763i)5-s + (1.84 + 3.19i)7-s − 1.91·8-s + (0.0747 + 0.129i)10-s + (1.97 + 3.42i)11-s + 4.06·13-s + (3.12 + 5.41i)14-s − 4.98·16-s + (0.586 − 1.01i)17-s + (3.26 − 2.89i)19-s + (0.0383 + 0.0664i)20-s + (3.34 + 5.80i)22-s − 3.83·23-s + ⋯
L(s)  = 1  + 1.19·2-s + 0.435·4-s + (0.0197 + 0.0341i)5-s + (0.698 + 1.20i)7-s − 0.676·8-s + (0.0236 + 0.0409i)10-s + (0.596 + 1.03i)11-s + 1.12·13-s + (0.836 + 1.44i)14-s − 1.24·16-s + (0.142 − 0.246i)17-s + (0.748 − 0.663i)19-s + (0.00858 + 0.0148i)20-s + (0.714 + 1.23i)22-s − 0.799·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.814 - 0.580i$
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: \((2,\ 513,\ (\ :1/2),\ 0.814 - 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50275 + 0.800063i\)
\(L(\frac12)\) \(\approx\) \(2.50275 + 0.800063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.26 + 2.89i)T \)
good2 \( 1 - 1.69T + 2T^{2} \)
5 \( 1 + (-0.0441 - 0.0763i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.84 - 3.19i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.97 - 3.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.06T + 13T^{2} \)
17 \( 1 + (-0.586 + 1.01i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + 3.83T + 23T^{2} \)
29 \( 1 + (-3.28 + 5.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.14 - 7.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.88T + 37T^{2} \)
41 \( 1 + (2.33 + 4.03i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 8.24T + 43T^{2} \)
47 \( 1 + (-2.21 + 3.84i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.62 - 6.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.505 - 0.876i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.61 - 2.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 2.90T + 67T^{2} \)
71 \( 1 + (-4.36 + 7.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.43 + 5.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.31T + 79T^{2} \)
83 \( 1 + (3.34 + 5.80i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.41 + 7.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.78T + 97T^{2} \)
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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

−11.44138735274629091385748952264, −10.17535933067236239615670759315, −9.005128204340670494197422125168, −8.506723618158433166868211587747, −7.04063714943304942762209738044, −6.07889828724392776448308621160, −5.21691899350870582677732800938, −4.43391155579426006537863436961, −3.25001832488311733922338274859, −1.96835877621698063393276989106, 1.30597137329743628448887840811, 3.47321228320147775276271195135, 3.85843974542933759009811594796, 5.08297967675171684738633463870, 5.95254127182703555431752803205, 6.89726228329005282946046976214, 8.094769934178254676619623354259, 8.897011008075174161560310095331, 10.13719203386569812832310219614, 11.17219973047770586540351089080

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