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Properties

Label 2-819-91.9-c1-0-11
Degree $2$
Conductor $819$
Sign $0.367 - 0.930i$
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  + (0.0340 + 0.0589i)2-s + (0.997 − 1.72i)4-s + (−1.52 + 2.64i)5-s + (−2.60 − 0.453i)7-s + 0.271·8-s − 0.208·10-s + 4.35·11-s + (−1.79 + 3.12i)13-s + (−0.0619 − 0.169i)14-s + (−1.98 − 3.44i)16-s + (−1.76 + 3.05i)17-s + 6.90·19-s + (3.05 + 5.28i)20-s + (0.148 + 0.256i)22-s + (1.66 + 2.88i)23-s + ⋯
L(s)  = 1  + (0.0240 + 0.0416i)2-s + (0.498 − 0.864i)4-s + (−0.684 + 1.18i)5-s + (−0.985 − 0.171i)7-s + 0.0961·8-s − 0.0658·10-s + 1.31·11-s + (−0.499 + 0.866i)13-s + (−0.0165 − 0.0451i)14-s + (−0.496 − 0.860i)16-s + (−0.427 + 0.741i)17-s + 1.58·19-s + (0.682 + 1.18i)20-s + (0.0316 + 0.0547i)22-s + (0.347 + 0.602i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.367 - 0.930i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.367 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06589 + 0.725012i\)
\(L(\frac12)\) \(\approx\) \(1.06589 + 0.725012i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 + 0.453i)T \)
13 \( 1 + (1.79 - 3.12i)T \)
good2 \( 1 + (-0.0340 - 0.0589i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.52 - 2.64i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
17 \( 1 + (1.76 - 3.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 6.90T + 19T^{2} \)
23 \( 1 + (-1.66 - 2.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.95 - 8.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.62 - 8.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0545 - 0.0944i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.76 - 3.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.844 + 1.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.28 - 2.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.65 + 4.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.77 + 6.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 4.87T + 61T^{2} \)
67 \( 1 - 0.680T + 67T^{2} \)
71 \( 1 + (-2.61 - 4.53i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.75 - 3.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.85 + 8.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.41T + 83T^{2} \)
89 \( 1 + (3.85 + 6.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.86 + 6.69i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.37467307426398810005389794992, −9.680305836769620435396841709642, −8.961965806968757438794927285904, −7.37029879776543769440502329661, −6.82892669650200699816987341010, −6.43023517758716407799076720225, −5.15413380478920425919214720322, −3.75698471645858965059414192780, −3.04345406758176921212306571544, −1.48699418714608180642156120355, 0.66457634493936205488864598850, 2.59079724979655038206422614439, 3.65095561444314718746781074692, 4.43492018011108568311360928521, 5.67734733718088908195354954070, 6.75837267232871166629272079239, 7.57620017151408344512772735186, 8.320283177563942108947277153034, 9.267872171455672719212761744211, 9.752776452589323738564182223341

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