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Properties

Label 2-819-91.74-c1-0-0
Degree $2$
Conductor $819$
Sign $-0.998 + 0.0519i$
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  − 2.28·2-s + 3.19·4-s + (1.46 + 2.54i)5-s + (0.102 − 2.64i)7-s − 2.73·8-s + (−3.34 − 5.80i)10-s + (−2.58 − 4.47i)11-s + (−0.364 + 3.58i)13-s + (−0.233 + 6.02i)14-s − 0.160·16-s − 5.05·17-s + (−1.12 + 1.95i)19-s + (4.69 + 8.13i)20-s + (5.89 + 10.2i)22-s − 5.23·23-s + ⋯
L(s)  = 1  − 1.61·2-s + 1.59·4-s + (0.656 + 1.13i)5-s + (0.0386 − 0.999i)7-s − 0.967·8-s + (−1.05 − 1.83i)10-s + (−0.779 − 1.34i)11-s + (−0.101 + 0.994i)13-s + (−0.0623 + 1.61i)14-s − 0.0400·16-s − 1.22·17-s + (−0.259 + 0.448i)19-s + (1.05 + 1.82i)20-s + (1.25 + 2.17i)22-s − 1.09·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.998 + 0.0519i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.998 + 0.0519i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00118018 - 0.0454088i\)
\(L(\frac12)\) \(\approx\) \(0.00118018 - 0.0454088i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.102 + 2.64i)T \)
13 \( 1 + (0.364 - 3.58i)T \)
good2 \( 1 + 2.28T + 2T^{2} \)
5 \( 1 + (-1.46 - 2.54i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.58 + 4.47i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
19 \( 1 + (1.12 - 1.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.23T + 23T^{2} \)
29 \( 1 + (0.216 - 0.375i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.34 - 2.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 + (0.269 - 0.466i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.66 + 8.07i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.87 - 8.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.377 - 0.653i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.64T + 59T^{2} \)
61 \( 1 + (3.47 - 6.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.68 - 11.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.90 + 6.76i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.94 - 13.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.79 + 13.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 - 4.00T + 89T^{2} \)
97 \( 1 + (-2.69 - 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.46768303384088592528399098314, −10.01818713324958215509451937177, −8.944273878940630087507510063009, −8.284879867804511521584088244946, −7.25747452625361658780446974865, −6.73678000627627893781067405863, −5.87245504760371500407365809027, −4.15252842032218187405407937626, −2.78152051244093891732122052620, −1.72595736927934022506279752350, 0.03524442634483266893619613886, 1.78517219229868751992989724584, 2.43568933957297199742350173669, 4.64166126253803575787733849962, 5.42385547199118551280855729786, 6.52427293609580677173507409459, 7.61921721366825474467102992486, 8.382359622743081049137852527311, 8.953794636220962194453725949179, 9.708969906084086960609907997956

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