Properties

Label 2-819-91.89-c1-0-21
Degree $2$
Conductor $819$
Sign $0.620 - 0.784i$
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  + (1.20 + 1.20i)2-s + 0.906i·4-s + (−0.363 − 1.35i)5-s + (0.864 + 2.50i)7-s + (1.31 − 1.31i)8-s + (1.19 − 2.07i)10-s + (−0.392 − 1.46i)11-s + (−1.37 + 3.33i)13-s + (−1.97 + 4.05i)14-s + 4.99·16-s + 5.17·17-s + (4.68 + 1.25i)19-s + (1.23 − 0.329i)20-s + (1.29 − 2.23i)22-s + 2.14i·23-s + ⋯
L(s)  = 1  + (0.852 + 0.852i)2-s + 0.453i·4-s + (−0.162 − 0.607i)5-s + (0.326 + 0.945i)7-s + (0.466 − 0.466i)8-s + (0.379 − 0.656i)10-s + (−0.118 − 0.441i)11-s + (−0.380 + 0.924i)13-s + (−0.527 + 1.08i)14-s + 1.24·16-s + 1.25·17-s + (1.07 + 0.288i)19-s + (0.275 − 0.0737i)20-s + (0.275 − 0.477i)22-s + 0.448i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.620 - 0.784i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.620 - 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35206 + 1.13834i\)
\(L(\frac12)\) \(\approx\) \(2.35206 + 1.13834i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.864 - 2.50i)T \)
13 \( 1 + (1.37 - 3.33i)T \)
good2 \( 1 + (-1.20 - 1.20i)T + 2iT^{2} \)
5 \( 1 + (0.363 + 1.35i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.392 + 1.46i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 5.17T + 17T^{2} \)
19 \( 1 + (-4.68 - 1.25i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 2.14iT - 23T^{2} \)
29 \( 1 + (-0.744 - 1.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.89 - 0.506i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.70 + 6.70i)T - 37iT^{2} \)
41 \( 1 + (6.34 + 1.70i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.27 + 4.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.20 - 1.66i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.87 + 3.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.98 - 5.98i)T + 59iT^{2} \)
61 \( 1 + (2.79 - 1.61i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.5 - 3.37i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.71 + 0.726i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.82 - 14.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.67 + 6.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.0684 - 0.0684i)T - 83iT^{2} \)
89 \( 1 + (7.89 + 7.89i)T + 89iT^{2} \)
97 \( 1 + (3.39 + 12.6i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.20927022068921724210645770556, −9.416856710088734719104653367820, −8.471536895546363403038827584807, −7.68000008577823344266235977103, −6.76388188496335612781332554967, −5.66376758904568653646432596103, −5.26610692477193760467499236233, −4.32688421189535684088885658323, −3.13145659520054238225547144576, −1.41265848604896794120249204339, 1.29264866205599529242006908011, 2.89555660535995926070799603915, 3.40353187890933147230203718484, 4.63496531776912512724393911374, 5.24834287567632916513678139758, 6.65233434029035580228912244761, 7.68878761847727884252365854241, 8.045588785400896517329127401263, 9.828956749728989738027606482558, 10.24761546421995183244400535638

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