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Properties

Label 2-819-91.6-c1-0-36
Degree $2$
Conductor $819$
Sign $0.939 + 0.342i$
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  + (−0.0473 − 0.176i)2-s + (1.70 − 0.983i)4-s + (2.80 + 2.80i)5-s + (1.70 − 2.02i)7-s + (−0.513 − 0.513i)8-s + (0.362 − 0.627i)10-s + (2.53 − 0.679i)11-s + (−1.37 − 3.33i)13-s + (−0.437 − 0.205i)14-s + (1.90 − 3.29i)16-s + (−1.43 − 2.48i)17-s + (−0.759 + 2.83i)19-s + (7.52 + 2.01i)20-s + (−0.240 − 0.415i)22-s + (−7.27 − 4.19i)23-s + ⋯
L(s)  = 1  + (−0.0334 − 0.124i)2-s + (0.851 − 0.491i)4-s + (1.25 + 1.25i)5-s + (0.645 − 0.764i)7-s + (−0.181 − 0.181i)8-s + (0.114 − 0.198i)10-s + (0.764 − 0.204i)11-s + (−0.380 − 0.924i)13-s + (−0.117 − 0.0550i)14-s + (0.475 − 0.822i)16-s + (−0.347 − 0.601i)17-s + (−0.174 + 0.650i)19-s + (1.68 + 0.450i)20-s + (−0.0511 − 0.0886i)22-s + (−1.51 − 0.875i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38049 - 0.420771i\)
\(L(\frac12)\) \(\approx\) \(2.38049 - 0.420771i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 2.02i)T \)
13 \( 1 + (1.37 + 3.33i)T \)
good2 \( 1 + (0.0473 + 0.176i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-2.80 - 2.80i)T + 5iT^{2} \)
11 \( 1 + (-2.53 + 0.679i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.43 + 2.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.759 - 2.83i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (7.27 + 4.19i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.66 - 2.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.75 - 6.75i)T + 31iT^{2} \)
37 \( 1 + (6.77 - 1.81i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.79 - 0.747i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.43 - 1.40i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.85 - 4.85i)T - 47iT^{2} \)
53 \( 1 - 5.43T + 53T^{2} \)
59 \( 1 + (0.00666 + 0.00178i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.65 - 3.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.10 - 7.84i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-14.5 - 3.91i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.321 - 0.321i)T - 73iT^{2} \)
79 \( 1 - 0.280T + 79T^{2} \)
83 \( 1 + (2.42 + 2.42i)T + 83iT^{2} \)
89 \( 1 + (-0.0536 - 0.200i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.197 - 0.736i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.14348285462162347874048105125, −9.939945668062349970022639282550, −8.454993377989692533084351651888, −7.32725554709328821363646763733, −6.64983691398787414234679008174, −6.04277137771321703522994034366, −5.01815297112549286814725364411, −3.43591835711441290038077158641, −2.42670211847030456992003489470, −1.43388942110090889251798599841, 1.81822164070149410221773473723, 2.12152210757949382915529245935, 4.02678831905137058257541061859, 5.04573550180205687722023567462, 5.99614267062004371910980780989, 6.59172446777538389263907707183, 7.904936341534629987998279377305, 8.637745334631430575903951053846, 9.321881502655777192226818485154, 10.07112677000519043427352549188

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