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Properties

Label 2-819-117.43-c1-0-43
Degree $2$
Conductor $819$
Sign $0.398 - 0.917i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
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  + (−1.46 + 0.844i)2-s + (1.40 + 1.01i)3-s + (0.426 − 0.738i)4-s + (−0.617 + 0.356i)5-s + (−2.90 − 0.307i)6-s − i·7-s − 1.93i·8-s + (0.922 + 2.85i)9-s + (0.602 − 1.04i)10-s + (3.08 − 1.77i)11-s + (1.35 − 0.600i)12-s + (3.60 + 0.133i)13-s + (0.844 + 1.46i)14-s + (−1.22 − 0.130i)15-s + (2.48 + 4.31i)16-s + (−3.21 − 5.57i)17-s + ⋯
L(s)  = 1  + (−1.03 + 0.597i)2-s + (0.808 + 0.588i)3-s + (0.213 − 0.369i)4-s + (−0.276 + 0.159i)5-s + (−1.18 − 0.125i)6-s − 0.377i·7-s − 0.684i·8-s + (0.307 + 0.951i)9-s + (0.190 − 0.329i)10-s + (0.929 − 0.536i)11-s + (0.389 − 0.173i)12-s + (0.999 + 0.0369i)13-s + (0.225 + 0.390i)14-s + (−0.317 − 0.0335i)15-s + (0.622 + 1.07i)16-s + (−0.780 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.398 - 0.917i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.398 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02009 + 0.669198i\)
\(L(\frac12)\) \(\approx\) \(1.02009 + 0.669198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.40 - 1.01i)T \)
7 \( 1 + iT \)
13 \( 1 + (-3.60 - 0.133i)T \)
good2 \( 1 + (1.46 - 0.844i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (0.617 - 0.356i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.08 + 1.77i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.21 + 5.57i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.89 + 2.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.46T + 23T^{2} \)
29 \( 1 + (-0.631 - 1.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.741 + 0.428i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.42 - 4.28i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.50iT - 41T^{2} \)
43 \( 1 + 0.398T + 43T^{2} \)
47 \( 1 + (5.12 + 2.95i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 13.2T + 53T^{2} \)
59 \( 1 + (-10.3 - 5.97i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 1.75T + 61T^{2} \)
67 \( 1 - 4.91iT - 67T^{2} \)
71 \( 1 + (2.27 - 1.31i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.62iT - 73T^{2} \)
79 \( 1 + (-7.90 + 13.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.668 - 0.386i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.958 + 0.553i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.95iT - 97T^{2} \)
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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

−10.01878853849820392762479288450, −9.069072429734138183512740594366, −9.036725007067404908199563122686, −7.932491539518690651223481888774, −7.21603300573661084189308637374, −6.47612631502586968051005161921, −4.95182634630271724683077381020, −3.83716152313420210339549820321, −3.06926932600741449601677496495, −1.08074521402871417830861067441, 1.12480382513583216877706512228, 1.98939488600507272583941750073, 3.26208683026890949614392179579, 4.36745231035238951289658728481, 5.94561818840363996549016272659, 6.81599367725216745657214887861, 8.082011131371109931491671604271, 8.355968389272691186923509906444, 9.308277954014537831909559212735, 9.699793216570653208984299145428

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