Properties

Label 2-819-91.81-c1-0-9
Degree $2$
Conductor $819$
Sign $-0.397 - 0.917i$
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  + (−0.952 + 1.65i)2-s + (−0.815 − 1.41i)4-s + (−0.736 − 1.27i)5-s + (−2.62 − 0.311i)7-s − 0.702·8-s + 2.80·10-s + 4.39·11-s + (2.69 + 2.39i)13-s + (3.01 − 4.03i)14-s + (2.30 − 3.98i)16-s + (−0.601 − 1.04i)17-s + 3.24·19-s + (−1.20 + 2.08i)20-s + (−4.18 + 7.25i)22-s + (−2.21 + 3.84i)23-s + ⋯
L(s)  = 1  + (−0.673 + 1.16i)2-s + (−0.407 − 0.706i)4-s + (−0.329 − 0.570i)5-s + (−0.993 − 0.117i)7-s − 0.248·8-s + 0.887·10-s + 1.32·11-s + (0.748 + 0.663i)13-s + (0.806 − 1.07i)14-s + (0.575 − 0.996i)16-s + (−0.145 − 0.252i)17-s + 0.743·19-s + (−0.268 + 0.465i)20-s + (−0.892 + 1.54i)22-s + (−0.462 + 0.801i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.397 - 0.917i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.397 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.476618 + 0.725701i\)
\(L(\frac12)\) \(\approx\) \(0.476618 + 0.725701i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.311i)T \)
13 \( 1 + (-2.69 - 2.39i)T \)
good2 \( 1 + (0.952 - 1.65i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.736 + 1.27i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 4.39T + 11T^{2} \)
17 \( 1 + (0.601 + 1.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.24T + 19T^{2} \)
23 \( 1 + (2.21 - 3.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0837 - 0.145i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.62 - 4.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.52 - 6.10i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.58 - 4.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0113 - 0.0197i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.84 - 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0708 - 0.122i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.67 + 4.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 4.13T + 67T^{2} \)
71 \( 1 + (4.98 - 8.63i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.62 - 13.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.387 + 0.670i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 + (-3.27 + 5.67i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.74 - 3.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.05041547047276745475168484499, −9.253614234796600407088166843702, −8.887502791359947743258396847621, −7.953077146126683345921337399628, −6.93645869249426987087631425922, −6.48354661788991328096886682507, −5.56368326323524998421291193744, −4.22389555974667048956509401093, −3.23915842275234660293756436133, −1.09419202127570462441083549754, 0.67812431496571179474109160294, 2.17773272805436445327163288912, 3.41122146961116322413241208648, 3.80981875509335662721218064036, 5.75541911703326375591752241278, 6.49981286715575383596665558437, 7.48152994197107069302390920903, 8.785365466180076060150044387182, 9.169172124374776067926450536518, 10.14959435380545066634671227775

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