Properties

Label 2-819-117.43-c1-0-44
Degree $2$
Conductor $819$
Sign $0.826 + 0.562i$
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  + (2.01 − 1.16i)2-s + (−1.69 + 0.363i)3-s + (1.71 − 2.96i)4-s + (−2.47 + 1.42i)5-s + (−2.99 + 2.70i)6-s + i·7-s − 3.31i·8-s + (2.73 − 1.23i)9-s + (−3.32 + 5.75i)10-s + (4.29 − 2.47i)11-s + (−1.82 + 5.64i)12-s + (3.40 + 1.18i)13-s + (1.16 + 2.01i)14-s + (3.66 − 3.31i)15-s + (−0.436 − 0.756i)16-s + (0.972 + 1.68i)17-s + ⋯
L(s)  = 1  + (1.42 − 0.823i)2-s + (−0.977 + 0.209i)3-s + (0.855 − 1.48i)4-s + (−1.10 + 0.638i)5-s + (−1.22 + 1.10i)6-s + 0.377i·7-s − 1.17i·8-s + (0.911 − 0.410i)9-s + (−1.05 + 1.82i)10-s + (1.29 − 0.747i)11-s + (−0.525 + 1.62i)12-s + (0.944 + 0.328i)13-s + (0.311 + 0.539i)14-s + (0.946 − 0.856i)15-s + (−0.109 − 0.189i)16-s + (0.235 + 0.408i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.826 + 0.562i$
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: \((2,\ 819,\ (\ :1/2),\ 0.826 + 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24310 - 0.690482i\)
\(L(\frac12)\) \(\approx\) \(2.24310 - 0.690482i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 - 0.363i)T \)
7 \( 1 - iT \)
13 \( 1 + (-3.40 - 1.18i)T \)
good2 \( 1 + (-2.01 + 1.16i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (2.47 - 1.42i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.29 + 2.47i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.972 - 1.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.59 + 2.65i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.75T + 23T^{2} \)
29 \( 1 + (0.552 + 0.957i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.87 + 3.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.52 - 0.877i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.825iT - 41T^{2} \)
43 \( 1 - 2.10T + 43T^{2} \)
47 \( 1 + (-3.71 - 2.14i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.62T + 53T^{2} \)
59 \( 1 + (-0.478 - 0.276i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 15.0T + 61T^{2} \)
67 \( 1 - 14.9iT - 67T^{2} \)
71 \( 1 + (8.29 - 4.78i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 + (-6.51 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.39 + 3.11i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.99 + 2.88i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.1iT - 97T^{2} \)
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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

−10.72183158082809472743357501365, −9.716015719482052779932525374585, −8.479530828285618326326475524403, −7.18897636753569778149933396089, −6.16757475275086416787942901173, −5.77525439974270174601426775269, −4.34604917008095966709609103358, −3.93451471277055884171743113429, −3.02378772362110914833066356308, −1.22950366749980461505480294914, 1.12879139405569203532707171799, 3.57968792615258296093453197808, 4.19949261437451856203649256761, 4.90338885878458034329997907650, 5.89436789247133052006042761471, 6.63247010738789988435866075736, 7.47075194087603628001728460690, 8.059726960873880841668480231794, 9.464801066008709889788875789356, 10.60732412214599717431404462695

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