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Properties

Label 2-819-91.41-c1-0-15
Degree $2$
Conductor $819$
Sign $0.456 + 0.889i$
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  + (−0.604 − 0.161i)2-s + (−1.39 − 0.804i)4-s + (−0.965 − 0.965i)5-s + (2.12 − 1.57i)7-s + (1.59 + 1.59i)8-s + (0.427 + 0.739i)10-s + (−1.26 + 4.72i)11-s + (1.35 + 3.34i)13-s + (−1.53 + 0.610i)14-s + (0.902 + 1.56i)16-s + (2.72 − 4.72i)17-s + (4.47 − 1.19i)19-s + (0.568 + 2.12i)20-s + (1.53 − 2.65i)22-s + (3.14 − 1.81i)23-s + ⋯
L(s)  = 1  + (−0.427 − 0.114i)2-s + (−0.696 − 0.402i)4-s + (−0.431 − 0.431i)5-s + (0.802 − 0.596i)7-s + (0.564 + 0.564i)8-s + (0.135 + 0.233i)10-s + (−0.382 + 1.42i)11-s + (0.374 + 0.927i)13-s + (−0.411 + 0.163i)14-s + (0.225 + 0.390i)16-s + (0.660 − 1.14i)17-s + (1.02 − 0.275i)19-s + (0.127 + 0.474i)20-s + (0.326 − 0.565i)22-s + (0.655 − 0.378i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.456 + 0.889i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (496, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.456 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.873942 - 0.533647i\)
\(L(\frac12)\) \(\approx\) \(0.873942 - 0.533647i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 1.57i)T \)
13 \( 1 + (-1.35 - 3.34i)T \)
good2 \( 1 + (0.604 + 0.161i)T + (1.73 + i)T^{2} \)
5 \( 1 + (0.965 + 0.965i)T + 5iT^{2} \)
11 \( 1 + (1.26 - 4.72i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.72 + 4.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.47 + 1.19i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.14 + 1.81i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.00 + 1.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.91 + 5.91i)T + 31iT^{2} \)
37 \( 1 + (2.84 - 10.6i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.08 + 4.04i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.669 + 0.386i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.65 + 5.65i)T - 47iT^{2} \)
53 \( 1 - 6.72T + 53T^{2} \)
59 \( 1 + (3.87 + 14.4i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.210 - 0.121i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.55 - 1.48i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.711 - 2.65i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.17 + 2.17i)T - 73iT^{2} \)
79 \( 1 - 8.38T + 79T^{2} \)
83 \( 1 + (-11.2 - 11.2i)T + 83iT^{2} \)
89 \( 1 + (3.25 + 0.872i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.46 + 2.53i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.867759774358556900425622735786, −9.403162826000980113846099508065, −8.417039650724850232234080168634, −7.64213933516907625705481995976, −6.94600810803820177757567301432, −5.24057300393381162473831171259, −4.78613934071016998484685325897, −3.91967483069423216943328471086, −2.02878450519809001392978018445, −0.77548274794261443281656269418, 1.13667068597878169235421662330, 3.15582014898764511755091660158, 3.72747518453160245544827394473, 5.28978127299323903129415945624, 5.74064860880074074739591015148, 7.40789736176911564881759294284, 7.87223450577522617200908724101, 8.656185191133257519414508317955, 9.236497845352846960109590508965, 10.65331052602093182378243056213

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