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Properties

Label 2-819-13.9-c1-0-28
Degree $2$
Conductor $819$
Sign $0.945 + 0.326i$
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  + (−0.866 + 1.5i)2-s + (−0.5 − 0.866i)4-s + 1.73·5-s + (−0.5 − 0.866i)7-s − 1.73·8-s + (−1.49 + 2.59i)10-s + (2.36 − 4.09i)11-s + (−1.59 − 3.23i)13-s + 1.73·14-s + (2.49 − 4.33i)16-s + (−2.13 − 3.69i)17-s + (−1 − 1.73i)19-s + (−0.866 − 1.50i)20-s + (4.09 + 7.09i)22-s + (0.633 − 1.09i)23-s + ⋯
L(s)  = 1  + (−0.612 + 1.06i)2-s + (−0.250 − 0.433i)4-s + 0.774·5-s + (−0.188 − 0.327i)7-s − 0.612·8-s + (−0.474 + 0.821i)10-s + (0.713 − 1.23i)11-s + (−0.443 − 0.896i)13-s + 0.462·14-s + (0.624 − 1.08i)16-s + (−0.517 − 0.896i)17-s + (−0.229 − 0.397i)19-s + (−0.193 − 0.335i)20-s + (0.873 + 1.51i)22-s + (0.132 − 0.228i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.945 + 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.937284 - 0.157330i\)
\(L(\frac12)\) \(\approx\) \(0.937284 - 0.157330i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (1.59 + 3.23i)T \)
good2 \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 + (-2.36 + 4.09i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.13 + 3.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.633 + 1.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.19T + 31T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.59 + 4.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.09 + 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.928T + 47T^{2} \)
53 \( 1 + 3.92T + 53T^{2} \)
59 \( 1 + (-5.36 - 9.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.59 - 13.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.09 - 3.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.19T + 73T^{2} \)
79 \( 1 - 5.80T + 79T^{2} \)
83 \( 1 + 8.19T + 83T^{2} \)
89 \( 1 + (0.464 - 0.803i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.19 - 12.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.901552061327683069220925880720, −9.073305713945012182256118836115, −8.624912167287123623486729788541, −7.45686856943018704577452960359, −6.88609703555297680844077115328, −5.89710491337784289447302709654, −5.36922599130360424501552094173, −3.72700525261561998724332904938, −2.54707369611749812212089553033, −0.55418257381289744539281443874, 1.71928227525545593051709426587, 2.13368217756690315240089812823, 3.60807782800730049140746860940, 4.74337901744352711374104959661, 6.10119069840932667582459461277, 6.64306575877115645279583854417, 7.990503372398797110362661502527, 9.070354656079010310914820272221, 9.685773088409898706865700511657, 9.955244959097730361749313219678

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