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Properties

Label 2-819-91.88-c1-0-6
Degree $2$
Conductor $819$
Sign $-0.721 - 0.692i$
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  + (1.19 + 0.689i)2-s + (−0.0491 − 0.0850i)4-s + (−0.697 + 0.402i)5-s + (−2.25 + 1.38i)7-s − 2.89i·8-s − 1.11·10-s + 5.27i·11-s + (−2.36 + 2.72i)13-s + (−3.64 + 0.0965i)14-s + (1.89 − 3.28i)16-s + (0.280 + 0.485i)17-s + 5.84i·19-s + (0.0685 + 0.0395i)20-s + (−3.63 + 6.29i)22-s + (0.802 − 1.38i)23-s + ⋯
L(s)  = 1  + (0.844 + 0.487i)2-s + (−0.0245 − 0.0425i)4-s + (−0.312 + 0.180i)5-s + (−0.852 + 0.522i)7-s − 1.02i·8-s − 0.351·10-s + 1.58i·11-s + (−0.656 + 0.754i)13-s + (−0.974 + 0.0257i)14-s + (0.474 − 0.821i)16-s + (0.0679 + 0.117i)17-s + 1.34i·19-s + (0.0153 + 0.00884i)20-s + (−0.774 + 1.34i)22-s + (0.167 − 0.289i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.721 - 0.692i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.721 - 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.480776 + 1.19423i\)
\(L(\frac12)\) \(\approx\) \(0.480776 + 1.19423i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 - 1.38i)T \)
13 \( 1 + (2.36 - 2.72i)T \)
good2 \( 1 + (-1.19 - 0.689i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.697 - 0.402i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 5.27iT - 11T^{2} \)
17 \( 1 + (-0.280 - 0.485i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 5.84iT - 19T^{2} \)
23 \( 1 + (-0.802 + 1.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.14 - 1.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.01 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.07 - 0.620i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.803 - 0.463i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.22 + 3.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.32 - 1.92i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.72 + 4.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.52 - 5.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 7.30T + 61T^{2} \)
67 \( 1 + 7.34iT - 67T^{2} \)
71 \( 1 + (-8.06 - 4.65i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.33 + 2.50i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.68 + 9.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.81iT - 83T^{2} \)
89 \( 1 + (4.33 + 2.50i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.22 - 5.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.27616474732527099403365250335, −9.761751662284558093739359083748, −9.057040101728140352217631427008, −7.60530106755226040279739651579, −6.96419506136193710485806612440, −6.16490037650124530870533080956, −5.23079894437603177845453005855, −4.32972577890139803532034758391, −3.46023647664782826628226323673, −1.96969470919098836496061739278, 0.46684567142951611100967267740, 2.71909741498232482848223373445, 3.34432197632435564587474580766, 4.30633764065418827575892752162, 5.30154353912989571246199344982, 6.19306940632906063252881301204, 7.35854783970455974569540149287, 8.242147764648846411787591500033, 9.032109258789037763792991867988, 10.08739223732980904260679434467

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