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Properties

Label 2-819-91.80-c1-0-14
Degree $2$
Conductor $819$
Sign $-0.375 - 0.926i$
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  + (0.449 + 1.67i)2-s + (−0.885 + 0.511i)4-s + (0.524 − 1.95i)5-s + (−0.616 + 2.57i)7-s + (1.20 + 1.20i)8-s + 3.52·10-s + (−2.56 − 2.56i)11-s + (1.78 + 3.13i)13-s + (−4.59 + 0.123i)14-s + (−2.49 + 4.32i)16-s + (0.752 + 1.30i)17-s + (2.92 + 2.92i)19-s + (0.536 + 2.00i)20-s + (3.15 − 5.45i)22-s + (3.21 + 1.85i)23-s + ⋯
L(s)  = 1  + (0.318 + 1.18i)2-s + (−0.442 + 0.255i)4-s + (0.234 − 0.875i)5-s + (−0.232 + 0.972i)7-s + (0.424 + 0.424i)8-s + 1.11·10-s + (−0.772 − 0.772i)11-s + (0.496 + 0.868i)13-s + (−1.22 + 0.0328i)14-s + (−0.624 + 1.08i)16-s + (0.182 + 0.316i)17-s + (0.670 + 0.670i)19-s + (0.119 + 0.447i)20-s + (0.671 − 1.16i)22-s + (0.670 + 0.386i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.375 - 0.926i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.375 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10108 + 1.63410i\)
\(L(\frac12)\) \(\approx\) \(1.10108 + 1.63410i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.616 - 2.57i)T \)
13 \( 1 + (-1.78 - 3.13i)T \)
good2 \( 1 + (-0.449 - 1.67i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.524 + 1.95i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.56 + 2.56i)T + 11iT^{2} \)
17 \( 1 + (-0.752 - 1.30i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.92 - 2.92i)T + 19iT^{2} \)
23 \( 1 + (-3.21 - 1.85i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.84 - 3.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.05 + 1.89i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.52 - 0.677i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.33 - 4.97i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.51 + 2.60i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.255 + 0.0684i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.63 + 4.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.24 + 0.334i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 10.6iT - 61T^{2} \)
67 \( 1 + (-9.48 + 9.48i)T - 67iT^{2} \)
71 \( 1 + (3.11 + 11.6i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.744 + 2.77i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.09 + 14.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.3 + 10.3i)T + 83iT^{2} \)
89 \( 1 + (-2.60 - 9.72i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.645 + 0.173i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.41667868997649029440811098628, −9.300318677453707892518140623746, −8.556509256681856230875763594408, −8.057867868670878588949400592466, −6.86368825529077865921123622997, −6.00420536860613649659550641480, −5.39979312042146305474633785931, −4.67130971579055911432343741738, −3.15709418485845570338354374263, −1.60375251299375685309564560763, 0.961480295579081593865600670989, 2.59184272108768955109826839385, 3.12485202927879302413695530211, 4.25925629110512304966127419821, 5.23866186552385672837084047892, 6.75022519864534997672762012014, 7.18969811016772693953596732450, 8.239290984959962729486950634908, 9.774928404427875316376429673571, 10.16165153185625379341922623399

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