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Properties

Label 2-819-91.23-c1-0-40
Degree $2$
Conductor $819$
Sign $-0.549 - 0.835i$
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  − 2.24i·2-s − 3.02·4-s + (−0.550 − 0.318i)5-s + (0.958 − 2.46i)7-s + 2.30i·8-s + (−0.712 + 1.23i)10-s + (−2.45 − 1.41i)11-s + (3.56 + 0.522i)13-s + (−5.52 − 2.14i)14-s − 0.893·16-s − 4.82·17-s + (1.33 − 0.772i)19-s + (1.66 + 0.962i)20-s + (−3.17 + 5.49i)22-s + 0.194·23-s + ⋯
L(s)  = 1  − 1.58i·2-s − 1.51·4-s + (−0.246 − 0.142i)5-s + (0.362 − 0.932i)7-s + 0.813i·8-s + (−0.225 + 0.390i)10-s + (−0.739 − 0.426i)11-s + (0.989 + 0.144i)13-s + (−1.47 − 0.574i)14-s − 0.223·16-s − 1.16·17-s + (0.306 − 0.177i)19-s + (0.372 + 0.215i)20-s + (−0.676 + 1.17i)22-s + 0.0405·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.549 - 0.835i$
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: \((2,\ 819,\ (\ :1/2),\ -0.549 - 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.445171 + 0.825679i\)
\(L(\frac12)\) \(\approx\) \(0.445171 + 0.825679i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.958 + 2.46i)T \)
13 \( 1 + (-3.56 - 0.522i)T \)
good2 \( 1 + 2.24iT - 2T^{2} \)
5 \( 1 + (0.550 + 0.318i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.45 + 1.41i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 + (-1.33 + 0.772i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.194T + 23T^{2} \)
29 \( 1 + (3.94 + 6.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.83 - 1.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.2iT - 37T^{2} \)
41 \( 1 + (-5.31 + 3.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.769 + 1.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.32 + 3.07i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.88 - 6.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.86iT - 59T^{2} \)
61 \( 1 + (-3.40 - 5.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.59 + 3.80i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.59 + 2.07i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.59 + 2.65i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.64 + 2.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.4iT - 83T^{2} \)
89 \( 1 + 2.22iT - 89T^{2} \)
97 \( 1 + (5.19 + 2.99i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.07175779272270604178603964088, −8.995525551838562938090577754263, −8.280887629954826930304268082501, −7.25127180554173122191280059440, −6.03511771759622279532028518085, −4.62037696794175913755500276177, −4.03320201277706214320589147787, −2.99260555270827912869411519813, −1.76470903996500264809588673585, −0.45723294132162739172344709269, 2.16435237217310658606103828968, 3.77086936570242794589059866705, 5.01197804394647106443385878776, 5.59758560197182687942018742584, 6.47330186238118245516435416888, 7.40625792564842616142460986029, 8.045185633224462979040702665181, 8.878808562844636369346157273828, 9.437414982029405727618404623713, 10.94875912330460498496998120590

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