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Properties

Label 2-819-91.73-c1-0-38
Degree $2$
Conductor $819$
Sign $0.407 + 0.913i$
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  + (0.186 − 0.694i)2-s + (1.28 + 0.741i)4-s + (1.87 + 0.501i)5-s + (−0.783 − 2.52i)7-s + (1.77 − 1.77i)8-s + (0.696 − 1.20i)10-s + (−0.825 − 3.08i)11-s + (0.846 − 3.50i)13-s + (−1.90 + 0.0737i)14-s + (0.582 + 1.00i)16-s + (0.254 − 0.440i)17-s + (−2.65 − 0.710i)19-s + (2.03 + 2.03i)20-s − 2.29·22-s + (−2.49 + 1.44i)23-s + ⋯
L(s)  = 1  + (0.131 − 0.491i)2-s + (0.642 + 0.370i)4-s + (0.836 + 0.224i)5-s + (−0.296 − 0.955i)7-s + (0.626 − 0.626i)8-s + (0.220 − 0.381i)10-s + (−0.248 − 0.929i)11-s + (0.234 − 0.972i)13-s + (−0.508 + 0.0196i)14-s + (0.145 + 0.252i)16-s + (0.0616 − 0.106i)17-s + (−0.608 − 0.162i)19-s + (0.454 + 0.454i)20-s − 0.488·22-s + (−0.520 + 0.300i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.407 + 0.913i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85149 - 1.20086i\)
\(L(\frac12)\) \(\approx\) \(1.85149 - 1.20086i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.783 + 2.52i)T \)
13 \( 1 + (-0.846 + 3.50i)T \)
good2 \( 1 + (-0.186 + 0.694i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.87 - 0.501i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.825 + 3.08i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.254 + 0.440i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.65 + 0.710i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.49 - 1.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.40T + 29T^{2} \)
31 \( 1 + (-0.827 - 3.08i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-9.40 - 2.51i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.34 + 5.34i)T - 41iT^{2} \)
43 \( 1 - 12.5iT - 43T^{2} \)
47 \( 1 + (2.88 - 10.7i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.42 + 5.93i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.74 - 1.00i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.51 - 3.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.56 + 1.75i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.90 - 1.90i)T + 71iT^{2} \)
73 \( 1 + (-0.252 + 0.0676i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.78 + 4.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.86 - 5.86i)T - 83iT^{2} \)
89 \( 1 + (-3.17 + 11.8i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.04 - 7.04i)T - 97iT^{2} \)
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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

−10.28184588631928264084947380978, −9.570092801065225931153203146286, −8.193115087150371378404081939759, −7.59226976813582271596364779490, −6.42004429820313712215335168076, −5.95438106988341301703733151640, −4.44991367247395944493683689179, −3.33987030551630647553595433170, −2.57951461218049500393691291385, −1.09913674590377546339731864172, 1.85554356994865440310373495487, 2.43364711397546811131420104978, 4.28280920563336175942486342311, 5.35231143651653083768875926324, 6.07010020565256235427084434167, 6.67699006918012324135574289977, 7.71503415000262712751006193549, 8.737737347188352083798693694618, 9.616771844597767492974418323306, 10.18933727435643289524376396494

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