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Properties

Label 2-819-91.20-c1-0-9
Degree $2$
Conductor $819$
Sign $-0.932 - 0.361i$
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  + (−1.96 + 0.525i)2-s + (1.84 − 1.06i)4-s + (−1.56 + 1.56i)5-s + (2.02 + 1.70i)7-s + (−0.187 + 0.187i)8-s + (2.25 − 3.90i)10-s + (1.16 + 4.33i)11-s + (−3.51 − 0.786i)13-s + (−4.87 − 2.27i)14-s + (−1.86 + 3.22i)16-s + (−1.31 − 2.27i)17-s + (6.01 + 1.61i)19-s + (−1.22 + 4.55i)20-s + (−4.56 − 7.90i)22-s + (4.58 + 2.64i)23-s + ⋯
L(s)  = 1  + (−1.38 + 0.371i)2-s + (0.922 − 0.532i)4-s + (−0.700 + 0.700i)5-s + (0.765 + 0.643i)7-s + (−0.0661 + 0.0661i)8-s + (0.712 − 1.23i)10-s + (0.350 + 1.30i)11-s + (−0.975 − 0.218i)13-s + (−1.30 − 0.609i)14-s + (−0.465 + 0.805i)16-s + (−0.318 − 0.552i)17-s + (1.37 + 0.369i)19-s + (−0.273 + 1.01i)20-s + (−0.972 − 1.68i)22-s + (0.956 + 0.552i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.932 - 0.361i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.932 - 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0956793 + 0.511477i\)
\(L(\frac12)\) \(\approx\) \(0.0956793 + 0.511477i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 - 1.70i)T \)
13 \( 1 + (3.51 + 0.786i)T \)
good2 \( 1 + (1.96 - 0.525i)T + (1.73 - i)T^{2} \)
5 \( 1 + (1.56 - 1.56i)T - 5iT^{2} \)
11 \( 1 + (-1.16 - 4.33i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.31 + 2.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.01 - 1.61i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.58 - 2.64i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.06 + 3.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.44 - 2.44i)T - 31iT^{2} \)
37 \( 1 + (-0.0290 - 0.108i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.03 + 7.58i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.47 - 3.16i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.82 - 5.82i)T + 47iT^{2} \)
53 \( 1 + 9.24T + 53T^{2} \)
59 \( 1 + (1.27 - 4.77i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.33 - 1.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.1 + 2.72i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.56 - 13.3i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (11.8 + 11.8i)T + 73iT^{2} \)
79 \( 1 + 13.3T + 79T^{2} \)
83 \( 1 + (5.15 - 5.15i)T - 83iT^{2} \)
89 \( 1 + (7.93 - 2.12i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.93 + 0.518i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.35725165040357492825323389704, −9.580407349988094255075378122198, −9.021339079722411686578362538306, −7.88610363837001122148469674914, −7.37911329951401556593270332406, −6.90254068939761620317227253293, −5.38612795720945100544141184868, −4.38519388765457560771172573309, −2.85999902652291578471310222740, −1.53602967991155055108151792838, 0.45502600509857589557490692397, 1.45276185797085109253815524925, 3.06997083692586604015632777304, 4.43582533021349952639736944910, 5.23388107858200117796973061956, 6.86079827386179233842209586020, 7.63856730198028044991847860176, 8.395909100913316222281470220248, 8.854555242373188610183795188136, 9.788477382209812521136509215141

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