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Properties

Label 2-819-91.41-c1-0-2
Degree $2$
Conductor $819$
Sign $-0.602 - 0.797i$
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  + (−1.61 − 0.433i)2-s + (0.700 + 0.404i)4-s + (1.42 + 1.42i)5-s + (0.234 + 2.63i)7-s + (1.41 + 1.41i)8-s + (−1.68 − 2.91i)10-s + (−0.254 + 0.948i)11-s + (−1.60 + 3.22i)13-s + (0.764 − 4.36i)14-s + (−2.48 − 4.29i)16-s + (−2.99 + 5.18i)17-s + (2.71 − 0.726i)19-s + (0.421 + 1.57i)20-s + (0.822 − 1.42i)22-s + (−2.58 + 1.49i)23-s + ⋯
L(s)  = 1  + (−1.14 − 0.306i)2-s + (0.350 + 0.202i)4-s + (0.635 + 0.635i)5-s + (0.0885 + 0.996i)7-s + (0.498 + 0.498i)8-s + (−0.532 − 0.922i)10-s + (−0.0766 + 0.285i)11-s + (−0.446 + 0.894i)13-s + (0.204 − 1.16i)14-s + (−0.620 − 1.07i)16-s + (−0.725 + 1.25i)17-s + (0.622 − 0.166i)19-s + (0.0941 + 0.351i)20-s + (0.175 − 0.303i)22-s + (−0.539 + 0.311i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.602 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.602 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.602 - 0.797i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (496, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.602 - 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245800 + 0.493902i\)
\(L(\frac12)\) \(\approx\) \(0.245800 + 0.493902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.234 - 2.63i)T \)
13 \( 1 + (1.60 - 3.22i)T \)
good2 \( 1 + (1.61 + 0.433i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.42 - 1.42i)T + 5iT^{2} \)
11 \( 1 + (0.254 - 0.948i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.99 - 5.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.71 + 0.726i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.58 - 1.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.65 + 6.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.11 + 6.11i)T + 31iT^{2} \)
37 \( 1 + (1.24 - 4.63i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.886 + 3.30i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.748 + 0.432i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.17 + 2.17i)T - 47iT^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 + (-0.131 - 0.491i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-11.2 - 6.51i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.827 + 0.221i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.01 - 11.2i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.03 - 1.03i)T - 73iT^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + (-1.23 - 1.23i)T + 83iT^{2} \)
89 \( 1 + (-7.75 - 2.07i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (12.0 - 3.23i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.23090952573608382532117534841, −9.691984953048124195784186597648, −9.031098414530109247834316046138, −8.229295862785791665124746806022, −7.29911163232607058601212079282, −6.26262525068356590876948760485, −5.40954848891556525653903772548, −4.13570357109498190261111585353, −2.38274333179179334010579691062, −1.89208844160022770839010633227, 0.39612196255012959679613141493, 1.60565643476544680788938226583, 3.36072801210585056580889766985, 4.68281094833554174134896921602, 5.50574626065318739244486510279, 6.90313552149247497790667244572, 7.44188739041021430030789203517, 8.306487053698351958187700050935, 9.218322454748122256215258723178, 9.659177605805982643182137412385

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