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Properties

Label 2-819-91.23-c1-0-26
Degree $2$
Conductor $819$
Sign $0.917 - 0.397i$
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  − 0.169i·2-s + 1.97·4-s + (2.65 + 1.53i)5-s + (0.886 + 2.49i)7-s − 0.673i·8-s + (0.259 − 0.450i)10-s + (−2.98 − 1.72i)11-s + (3.09 − 1.85i)13-s + (0.422 − 0.150i)14-s + 3.82·16-s + 4.67·17-s + (−2.95 + 1.70i)19-s + (5.23 + 3.02i)20-s + (−0.291 + 0.505i)22-s − 8.27·23-s + ⋯
L(s)  = 1  − 0.119i·2-s + 0.985·4-s + (1.18 + 0.685i)5-s + (0.335 + 0.942i)7-s − 0.238i·8-s + (0.0821 − 0.142i)10-s + (−0.899 − 0.519i)11-s + (0.857 − 0.514i)13-s + (0.112 − 0.0401i)14-s + 0.957·16-s + 1.13·17-s + (−0.677 + 0.391i)19-s + (1.16 + 0.675i)20-s + (−0.0622 + 0.107i)22-s − 1.72·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.917 - 0.397i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (478, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.917 - 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37029 + 0.491608i\)
\(L(\frac12)\) \(\approx\) \(2.37029 + 0.491608i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.886 - 2.49i)T \)
13 \( 1 + (-3.09 + 1.85i)T \)
good2 \( 1 + 0.169iT - 2T^{2} \)
5 \( 1 + (-2.65 - 1.53i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.98 + 1.72i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.67T + 17T^{2} \)
19 \( 1 + (2.95 - 1.70i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.27T + 23T^{2} \)
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 \( 1 - 7.63iT - 37T^{2} \)
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 \( 1 + 11.6iT - 59T^{2} \)
61 \( 1 + (-0.823 - 1.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.31 + 1.33i)T + (33.5 + 58.0i)T^{2} \)
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 \( 1 + 1.41iT - 89T^{2} \)
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

−10.21044742447496069525088285188, −9.878785491229761863353498333554, −8.352979238006200346596826759323, −7.917320131924168067155577389928, −6.51797558403764387556692535900, −5.91746934597052293596438961381, −5.43955057811477095655982378502, −3.47511900860085787794640182191, −2.54181585368987249784441668776, −1.75145403453306226807260517530, 1.39879049003188516015409451048, 2.22592295334092913639624731615, 3.74935720389778107938069433785, 5.00954254047873807263981403414, 5.83936881921771645468744960167, 6.63255739675614422841047747786, 7.64036146582570297068519962907, 8.307459949568793184525031508317, 9.531998526841929351395029137160, 10.25405656672338833707077873744

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