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Properties

Label 2-819-91.81-c1-0-13
Degree $2$
Conductor $819$
Sign $0.803 - 0.595i$
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  + (0.5 − 0.866i)2-s + (0.500 + 0.866i)4-s + (1.5 + 2.59i)5-s + (−2.5 − 0.866i)7-s + 3·8-s + 3·10-s + 3·11-s + (−1 + 3.46i)13-s + (−2 + 1.73i)14-s + (0.500 − 0.866i)16-s + (−1 − 1.73i)17-s − 19-s + (−1.49 + 2.59i)20-s + (1.5 − 2.59i)22-s + (−2 + 3.46i)25-s + (2.49 + 2.59i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.944 − 0.327i)7-s + 1.06·8-s + 0.948·10-s + 0.904·11-s + (−0.277 + 0.960i)13-s + (−0.534 + 0.462i)14-s + (0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s − 0.229·19-s + (−0.335 + 0.580i)20-s + (0.319 − 0.553i)22-s + (−0.400 + 0.692i)25-s + (0.490 + 0.509i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.803 - 0.595i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05866 + 0.679393i\)
\(L(\frac12)\) \(\approx\) \(2.05866 + 0.679393i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + (-6.5 + 11.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.53740879796961421324587228093, −9.655047094514461952573233080019, −8.895456523111674092073921309165, −7.40281787909318073610684306078, −6.77719618186618218925323984773, −6.28640541746606310430207161269, −4.66343261007827102154211271454, −3.60146969727530470879276351785, −2.88371981609774629833788006898, −1.81727665223344909936360619514, 1.01105507972503611508071023563, 2.37982520516258210570106036149, 4.01518403068575725410340197495, 5.01525889891267634984076146382, 5.91922621625638921535247893916, 6.28927169178004838764085774679, 7.43710925856592540429954935178, 8.504518780505942534690225576577, 9.377756947538039223846623107417, 9.932464615620533737295198628603

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