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Properties

Label 2-819-91.30-c1-0-8
Degree $2$
Conductor $819$
Sign $0.121 - 0.992i$
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  + (0.146 − 0.0847i)2-s + (−0.985 + 1.70i)4-s + (−2.65 − 1.53i)5-s + (2.60 + 0.478i)7-s + 0.673i·8-s − 0.519·10-s − 3.44i·11-s + (3.09 + 1.85i)13-s + (0.422 − 0.150i)14-s + (−1.91 − 3.31i)16-s + (−2.33 + 4.04i)17-s + 3.40i·19-s + (5.23 − 3.02i)20-s + (−0.291 − 0.505i)22-s + (4.13 + 7.16i)23-s + ⋯
L(s)  = 1  + (0.103 − 0.0599i)2-s + (−0.492 + 0.853i)4-s + (−1.18 − 0.685i)5-s + (0.983 + 0.180i)7-s + 0.238i·8-s − 0.164·10-s − 1.03i·11-s + (0.857 + 0.514i)13-s + (0.112 − 0.0401i)14-s + (−0.478 − 0.828i)16-s + (−0.566 + 0.982i)17-s + 0.782i·19-s + (1.16 − 0.675i)20-s + (−0.0622 − 0.107i)22-s + (0.862 + 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.121 - 0.992i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.121 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796983 + 0.705149i\)
\(L(\frac12)\) \(\approx\) \(0.796983 + 0.705149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 0.478i)T \)
13 \( 1 + (-3.09 - 1.85i)T \)
good2 \( 1 + (-0.146 + 0.0847i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (2.65 + 1.53i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.44iT - 11T^{2} \)
17 \( 1 + (2.33 - 4.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 3.40iT - 19T^{2} \)
23 \( 1 + (-4.13 - 7.16i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.96 - 3.40i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.45 - 0.838i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.61 - 3.81i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.02 + 3.47i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.78 - 8.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-10.9 - 6.30i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.35 - 11.0i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.1 + 5.84i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 1.64T + 61T^{2} \)
67 \( 1 + 2.67iT - 67T^{2} \)
71 \( 1 + (-10.5 + 6.07i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.11 - 2.37i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.12 + 1.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.62iT - 83T^{2} \)
89 \( 1 + (-1.22 + 0.707i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.53 - 3.77i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.86213926393252230106363472740, −9.031277691742506588500591030790, −8.719309511052183528984020276765, −8.014189804681288293993821641402, −7.37970938664507643841514841167, −5.88283837436244500253254890246, −4.84682759508881106286763859854, −3.98707437751984715146943808899, −3.36348855513262338123735990606, −1.39914882799187258650854259138, 0.57507814410877862382678292089, 2.29357666786739991085024000724, 3.85126623522202975507207936354, 4.60217767070668656896690583720, 5.39553935067697414175152030729, 6.83870933312264230702357910586, 7.26774723028605676438985664121, 8.420882527150431926518811976234, 9.057530575002070771469650703774, 10.31385771945801778876004247832

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