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Properties

Label 2-819-91.76-c1-0-37
Degree $2$
Conductor $819$
Sign $-0.331 + 0.943i$
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  + (0.357 − 1.33i)2-s + (0.0814 + 0.0470i)4-s + (2.02 − 2.02i)5-s + (1.39 − 2.25i)7-s + (2.04 − 2.04i)8-s + (−1.98 − 3.43i)10-s + (−1.37 − 0.369i)11-s + (3.54 − 0.634i)13-s + (−2.50 − 2.65i)14-s + (−1.90 − 3.29i)16-s + (−2.09 + 3.63i)17-s + (1.59 + 5.95i)19-s + (0.260 − 0.0698i)20-s + (−0.985 + 1.70i)22-s + (−6.77 + 3.91i)23-s + ⋯
L(s)  = 1  + (0.252 − 0.942i)2-s + (0.0407 + 0.0235i)4-s + (0.907 − 0.907i)5-s + (0.525 − 0.850i)7-s + (0.722 − 0.722i)8-s + (−0.626 − 1.08i)10-s + (−0.415 − 0.111i)11-s + (0.984 − 0.176i)13-s + (−0.669 − 0.710i)14-s + (−0.475 − 0.823i)16-s + (−0.509 + 0.881i)17-s + (0.366 + 1.36i)19-s + (0.0582 − 0.0156i)20-s + (−0.210 + 0.364i)22-s + (−1.41 + 0.815i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.331 + 0.943i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42705 - 2.01426i\)
\(L(\frac12)\) \(\approx\) \(1.42705 - 2.01426i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.39 + 2.25i)T \)
13 \( 1 + (-3.54 + 0.634i)T \)
good2 \( 1 + (-0.357 + 1.33i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-2.02 + 2.02i)T - 5iT^{2} \)
11 \( 1 + (1.37 + 0.369i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.09 - 3.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.59 - 5.95i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (6.77 - 3.91i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.441 + 0.764i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.648 - 0.648i)T - 31iT^{2} \)
37 \( 1 + (7.19 + 1.92i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-11.4 - 3.07i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.809 - 0.467i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.20 + 2.20i)T + 47iT^{2} \)
53 \( 1 + 2.52T + 53T^{2} \)
59 \( 1 + (5.65 - 1.51i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.0739 + 0.0427i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.266 + 0.995i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.79 - 0.750i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.01 - 2.01i)T + 73iT^{2} \)
79 \( 1 - 9.43T + 79T^{2} \)
83 \( 1 + (1.54 - 1.54i)T - 83iT^{2} \)
89 \( 1 + (1.27 - 4.75i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.37 + 8.87i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.23937704032301071524427263216, −9.404290189003386136417301643345, −8.253219901736844476468718309346, −7.63661942061694642997757970764, −6.26949630128291929364808814145, −5.46140039510287669860120979085, −4.24402927756110797098100576455, −3.57159322750844284081821196543, −1.92939480103059974064808125060, −1.30191963749431742943507361806, 1.99376769891727893132497382846, 2.74415620210298439803072812150, 4.55693759536689554749529429792, 5.47371159740664565922112665251, 6.18449459905238499013507700437, 6.81105648977574040050406464247, 7.74080220477976894877488346463, 8.688385024251537065909006234068, 9.528047148862916034729235175055, 10.69339307980679335868229536791

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