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Properties

Label 2-819-91.20-c1-0-31
Degree $2$
Conductor $819$
Sign $0.914 + 0.404i$
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  + (0.892 − 0.239i)2-s + (−0.993 + 0.573i)4-s + (2.80 − 2.80i)5-s + (2.12 + 1.57i)7-s + (−2.05 + 2.05i)8-s + (1.82 − 3.16i)10-s + (−0.544 − 2.03i)11-s + (3.41 + 1.16i)13-s + (2.27 + 0.893i)14-s + (−0.195 + 0.339i)16-s + (1.58 + 2.74i)17-s + (1.12 + 0.302i)19-s + (−1.17 + 4.38i)20-s + (−0.972 − 1.68i)22-s + (−3.26 − 1.88i)23-s + ⋯
L(s)  = 1  + (0.630 − 0.169i)2-s + (−0.496 + 0.286i)4-s + (1.25 − 1.25i)5-s + (0.804 + 0.594i)7-s + (−0.726 + 0.726i)8-s + (0.578 − 1.00i)10-s + (−0.164 − 0.613i)11-s + (0.946 + 0.324i)13-s + (0.607 + 0.238i)14-s + (−0.0489 + 0.0847i)16-s + (0.384 + 0.666i)17-s + (0.258 + 0.0693i)19-s + (−0.262 + 0.980i)20-s + (−0.207 − 0.359i)22-s + (−0.680 − 0.392i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.914 + 0.404i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.914 + 0.404i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42556 - 0.513009i\)
\(L(\frac12)\) \(\approx\) \(2.42556 - 0.513009i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.41 - 1.16i)T \)
good2 \( 1 + (-0.892 + 0.239i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-2.80 + 2.80i)T - 5iT^{2} \)
11 \( 1 + (0.544 + 2.03i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.58 - 2.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.12 - 0.302i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.26 + 1.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.584 + 1.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.57 + 3.57i)T - 31iT^{2} \)
37 \( 1 + (-1.14 - 4.26i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.85 + 6.93i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.91 - 1.10i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.21 - 8.21i)T + 47iT^{2} \)
53 \( 1 + 4.89T + 53T^{2} \)
59 \( 1 + (0.0633 - 0.236i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-10.7 + 6.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.95 - 2.66i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.60 - 5.98i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.47 + 3.47i)T + 73iT^{2} \)
79 \( 1 + 2.69T + 79T^{2} \)
83 \( 1 + (3.31 - 3.31i)T - 83iT^{2} \)
89 \( 1 + (6.71 - 1.80i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (15.8 + 4.24i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.04963392520324222576413026880, −9.162100204757576755520280493613, −8.516352023649539923683058539614, −8.097428896479149269390544198415, −6.06424075041850931064614852862, −5.71560128502005383344848725375, −4.83870778625831266818259918940, −4.00497752287683616349821124905, −2.52261490337620529605160162694, −1.31335993213533559610853059062, 1.45504945403216018455616768591, 2.87025635511822649837105756804, 3.95214582053241901074251052948, 5.10911355380450322520300730043, 5.78999000144510839198405043246, 6.66529497293858176781328329057, 7.45681851015698534736836140863, 8.673177061580330977371551763369, 9.771781303805827448315039131828, 10.18841228694410426796605091461

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