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Properties

Label 2-513-171.49-c1-0-2
Degree $2$
Conductor $513$
Sign $0.248 - 0.968i$
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-s − 4-s + (1.5 + 2.59i)5-s + (−0.5 − 0.866i)7-s − 3·8-s + (1.5 + 2.59i)10-s + (2.5 + 4.33i)11-s + 2·13-s + (−0.5 − 0.866i)14-s − 16-s + (−2.5 + 4.33i)17-s + (−4 + 1.73i)19-s + (−1.5 − 2.59i)20-s + (2.5 + 4.33i)22-s + 8·23-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.5·4-s + (0.670 + 1.16i)5-s + (−0.188 − 0.327i)7-s − 1.06·8-s + (0.474 + 0.821i)10-s + (0.753 + 1.30i)11-s + 0.554·13-s + (−0.133 − 0.231i)14-s − 0.250·16-s + (−0.606 + 1.05i)17-s + (−0.917 + 0.397i)19-s + (−0.335 − 0.580i)20-s + (0.533 + 0.923i)22-s + 1.66·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.248 - 0.968i$
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: \((2,\ 513,\ (\ :1/2),\ 0.248 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40146 + 1.08726i\)
\(L(\frac12)\) \(\approx\) \(1.40146 + 1.08726i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - 1.73i)T \)
good2 \( 1 - T + 2T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.5 - 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.88038570025937315973689881314, −10.36159101781948981459378157298, −9.352890509838825044329121836570, −8.590614313617980126603795270848, −6.95781555306581834019505513278, −6.57030618555972615084422493908, −5.46301976718831207749452064709, −4.24844431772037051666896848580, −3.44325456370739893219454591961, −2.01474078976988284388639713737, 0.908231246757982693412524793332, 2.85709201888239440729044740170, 4.11292377461692986495831951384, 5.06450152971063791987442076242, 5.79137646571029282900666616071, 6.67081282037641343310408089975, 8.454700749589732818573235484125, 9.032124461165936428124012183113, 9.362046566952785027380932478605, 10.91384935628381307222436404766

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