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Properties

Label 2-819-117.103-c1-0-2
Degree $2$
Conductor $819$
Sign $-0.751 - 0.660i$
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  + (−1.30 + 0.755i)2-s + (−1.57 + 0.714i)3-s + (0.141 − 0.244i)4-s + (−3.48 − 2.01i)5-s + (1.52 − 2.12i)6-s + (−0.866 + 0.5i)7-s − 2.59i·8-s + (1.98 − 2.25i)9-s + 6.07·10-s + (−0.727 + 0.420i)11-s + (−0.0481 + 0.486i)12-s + (3.38 − 1.23i)13-s + (0.755 − 1.30i)14-s + (6.93 + 0.686i)15-s + (2.24 + 3.88i)16-s − 2.30·17-s + ⋯
L(s)  = 1  + (−0.925 + 0.534i)2-s + (−0.911 + 0.412i)3-s + (0.0705 − 0.122i)4-s + (−1.55 − 0.899i)5-s + (0.622 − 0.868i)6-s + (−0.327 + 0.188i)7-s − 0.917i·8-s + (0.660 − 0.751i)9-s + 1.92·10-s + (−0.219 + 0.126i)11-s + (−0.0138 + 0.140i)12-s + (0.939 − 0.341i)13-s + (0.201 − 0.349i)14-s + (1.79 + 0.177i)15-s + (0.560 + 0.970i)16-s − 0.559·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.751 - 0.660i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.751 - 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0440121 + 0.116759i\)
\(L(\frac12)\) \(\approx\) \(0.0440121 + 0.116759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.57 - 0.714i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-3.38 + 1.23i)T \)
good2 \( 1 + (1.30 - 0.755i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (3.48 + 2.01i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.727 - 0.420i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.30T + 17T^{2} \)
19 \( 1 + 6.65iT - 19T^{2} \)
23 \( 1 + (0.0775 - 0.134i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.79 + 3.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.73 + 3.88i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.83iT - 37T^{2} \)
41 \( 1 + (1.45 + 0.841i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.84 + 3.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.93 + 2.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.68T + 53T^{2} \)
59 \( 1 + (10.1 + 5.86i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.56 - 4.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.17 - 1.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.96iT - 71T^{2} \)
73 \( 1 - 9.47iT - 73T^{2} \)
79 \( 1 + (2.54 + 4.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.6 + 6.12i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.2iT - 89T^{2} \)
97 \( 1 + (9.08 - 5.24i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.57267612291482882105534676458, −9.381160133809691268730920169348, −8.896141068196698383859563986284, −8.077710735637778241362683808174, −7.26139717497153784050796823163, −6.46337907172386981640215189206, −5.22106692115784717206667593740, −4.27104316327694700515381497285, −3.54413087026872169008400489025, −0.790740999514787831888003082576, 0.15202894674600498011173042391, 1.68061854660873863636630102502, 3.30369608749556639765861627914, 4.29867728111019085397613094919, 5.65436542676913199490102386620, 6.61542038799117941375083370083, 7.51357609805137749266919258318, 8.099582587611480756418081325383, 9.089310981599971995166391122850, 10.30784131044040759457189268440

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