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Properties

Label 2-819-91.74-c1-0-29
Degree $2$
Conductor $819$
Sign $-0.269 + 0.963i$
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  − 0.883·2-s − 1.21·4-s + (−0.803 − 1.39i)5-s + (0.969 − 2.46i)7-s + 2.84·8-s + (0.709 + 1.22i)10-s + (2.72 + 4.71i)11-s + (−3.59 − 0.322i)13-s + (−0.855 + 2.17i)14-s − 0.0728·16-s + 4.79·17-s + (2.78 − 4.82i)19-s + (0.979 + 1.69i)20-s + (−2.40 − 4.16i)22-s − 7.52·23-s + ⋯
L(s)  = 1  − 0.624·2-s − 0.609·4-s + (−0.359 − 0.622i)5-s + (0.366 − 0.930i)7-s + 1.00·8-s + (0.224 + 0.388i)10-s + (0.821 + 1.42i)11-s + (−0.995 − 0.0894i)13-s + (−0.228 + 0.581i)14-s − 0.0182·16-s + 1.16·17-s + (0.638 − 1.10i)19-s + (0.219 + 0.379i)20-s + (−0.513 − 0.888i)22-s − 1.56·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.269 + 0.963i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.269 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.437687 - 0.576773i\)
\(L(\frac12)\) \(\approx\) \(0.437687 - 0.576773i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.969 + 2.46i)T \)
13 \( 1 + (3.59 + 0.322i)T \)
good2 \( 1 + 0.883T + 2T^{2} \)
5 \( 1 + (0.803 + 1.39i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.72 - 4.71i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.79T + 17T^{2} \)
19 \( 1 + (-2.78 + 4.82i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.52T + 23T^{2} \)
29 \( 1 + (-0.471 + 0.816i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.198 - 0.344i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.86T + 37T^{2} \)
41 \( 1 + (-2.37 + 4.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.30 + 9.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.143 + 0.247i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0573 - 0.0993i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.23T + 59T^{2} \)
61 \( 1 + (-5.19 + 8.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.38 + 4.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.93 + 12.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.63 - 6.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.55 + 4.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.73T + 83T^{2} \)
89 \( 1 + 4.28T + 89T^{2} \)
97 \( 1 + (-0.544 - 0.943i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.949071345352552204951997728672, −9.247682672002424567595607709183, −8.275215281208980845511024267668, −7.52423704575819031908847653699, −6.97384593713557159469734161101, −5.21222848608086088772438295577, −4.56425780435600383563637606009, −3.80745405040163560334215584601, −1.79812400917317567479868309621, −0.50722712988369440400906556064, 1.38692494384442310899113769508, 3.04847511194694640814365508312, 3.99326579611781708729764452670, 5.31889214099393906406658125636, 6.02212643428887298608145821199, 7.35942342363354751569072838615, 8.114025319983455230579104979204, 8.684505668747911965908393337963, 9.716759182373987528165565062290, 10.16737611483623907354373368280

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