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Properties

Label 2-819-91.4-c1-0-11
Degree $2$
Conductor $819$
Sign $0.945 + 0.325i$
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  − 2.18i·2-s − 2.79·4-s + (−1.5 + 0.866i)5-s + (2.29 + 1.32i)7-s + 1.73i·8-s + (1.89 + 3.28i)10-s + (−3 + 1.73i)11-s + (−1 + 3.46i)13-s + (2.89 − 5.01i)14-s − 1.79·16-s − 17-s + (4.58 + 2.64i)19-s + (4.18 − 2.41i)20-s + (3.79 + 6.56i)22-s + 8.58·23-s + ⋯
L(s)  = 1  − 1.54i·2-s − 1.39·4-s + (−0.670 + 0.387i)5-s + (0.866 + 0.499i)7-s + 0.612i·8-s + (0.599 + 1.03i)10-s + (−0.904 + 0.522i)11-s + (−0.277 + 0.960i)13-s + (0.773 − 1.34i)14-s − 0.447·16-s − 0.242·17-s + (1.05 + 0.606i)19-s + (0.936 − 0.540i)20-s + (0.808 + 1.40i)22-s + 1.78·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.945 + 0.325i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.945 + 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12280 - 0.187878i\)
\(L(\frac12)\) \(\approx\) \(1.12280 - 0.187878i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.29 - 1.32i)T \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 + 2.18iT - 2T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 + (-4.58 - 2.64i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.58T + 23T^{2} \)
29 \( 1 + (3.5 - 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.29 - 3.05i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.02iT - 37T^{2} \)
41 \( 1 + (-3.08 - 1.77i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.29 + 3.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.708 + 0.409i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.08 - 5.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.28iT - 59T^{2} \)
61 \( 1 + (-2.58 + 4.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.1 - 7.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.87 + 2.23i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.708 - 1.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 - 15.5iT - 89T^{2} \)
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

−10.54768205345591201577896735424, −9.455618469872235971114996000307, −8.865947711921103131378756875200, −7.68378962984424336214378756670, −7.00233508631424995119038833384, −5.29182888887573041405004148736, −4.58459826465688741655130203172, −3.46330405760525547649910581817, −2.53637715545144850508877551089, −1.44842099583519400398154705870, 0.60727476220761322079540109663, 2.91183315498519776141480229563, 4.47337038036019021751706779411, 5.03150211977101162849352580814, 5.85981300806408103781013948009, 7.07212731291390002389565129840, 7.78959324828657046255901358883, 8.125508813466311687285242644371, 9.003548173005947204779537982050, 10.15273854493555400219369288210

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