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Properties

Label 2-819-13.3-c1-0-24
Degree $2$
Conductor $819$
Sign $0.903 + 0.427i$
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  + (0.115 + 0.200i)2-s + (0.973 − 1.68i)4-s + 2.23·5-s + (0.5 − 0.866i)7-s + 0.913·8-s + (0.258 + 0.447i)10-s + (1.66 + 2.87i)11-s + (3.40 − 1.19i)13-s + 0.231·14-s + (−1.84 − 3.18i)16-s + (−0.687 + 1.19i)17-s + (−1.61 + 2.80i)19-s + (2.17 − 3.76i)20-s + (−0.384 + 0.665i)22-s + (0.419 + 0.726i)23-s + ⋯
L(s)  = 1  + (0.0817 + 0.141i)2-s + (0.486 − 0.842i)4-s + 0.997·5-s + (0.188 − 0.327i)7-s + 0.322·8-s + (0.0816 + 0.141i)10-s + (0.500 + 0.867i)11-s + (0.943 − 0.330i)13-s + 0.0618·14-s + (−0.460 − 0.797i)16-s + (−0.166 + 0.288i)17-s + (−0.371 + 0.642i)19-s + (0.485 − 0.841i)20-s + (−0.0819 + 0.141i)22-s + (0.0874 + 0.151i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.903 + 0.427i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.903 + 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24588 - 0.504823i\)
\(L(\frac12)\) \(\approx\) \(2.24588 - 0.504823i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-3.40 + 1.19i)T \)
good2 \( 1 + (-0.115 - 0.200i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
11 \( 1 + (-1.66 - 2.87i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.687 - 1.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.61 - 2.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.419 - 0.726i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.303 + 0.525i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.71T + 31T^{2} \)
37 \( 1 + (0.776 + 1.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.58 + 7.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.615 - 1.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.62T + 47T^{2} \)
53 \( 1 + 8.39T + 53T^{2} \)
59 \( 1 + (-4.41 + 7.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.73 - 4.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.09 - 8.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.60 - 4.51i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.96T + 73T^{2} \)
79 \( 1 - 6.45T + 79T^{2} \)
83 \( 1 - 4.64T + 83T^{2} \)
89 \( 1 + (-4.56 - 7.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.67 + 13.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.17381452761675557113253216157, −9.568904623775564918435935560416, −8.569709729765463292274722519891, −7.41067809307786975792646982800, −6.50113104510210448499844567322, −5.91261349111171978170090492364, −5.01324396248244041584596615679, −3.83546130091279791845836906343, −2.17872606804282345515574259826, −1.36560884520607906042953561646, 1.59049622045341954304734550592, 2.70619079108363743403151691882, 3.70711969487671020366548681278, 4.92057519829078949472155369007, 6.20545824693087430476595341921, 6.54780366700938742341383869378, 7.84233082640887675007859122117, 8.698567268735845939017931840460, 9.261858125157863425640120101678, 10.43985965991486438248870351683

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