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Properties

Label 2-819-7.2-c1-0-28
Degree $2$
Conductor $819$
Sign $-0.605 + 0.795i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.190 + 0.330i)2-s + (0.927 + 1.60i)4-s + (−1.11 + 1.93i)5-s + (−2 − 1.73i)7-s − 1.47·8-s + (−0.427 − 0.739i)10-s + (−1.5 − 2.59i)11-s − 13-s + (0.954 − 0.330i)14-s + (−1.57 + 2.72i)16-s + (−3.73 − 6.47i)17-s + (−1.5 + 2.59i)19-s − 4.14·20-s + 1.14·22-s + (−1.88 + 3.25i)23-s + ⋯
L(s)  = 1  + (−0.135 + 0.233i)2-s + (0.463 + 0.802i)4-s + (−0.499 + 0.866i)5-s + (−0.755 − 0.654i)7-s − 0.520·8-s + (−0.135 − 0.233i)10-s + (−0.452 − 0.783i)11-s − 0.277·13-s + (0.255 − 0.0884i)14-s + (−0.393 + 0.681i)16-s + (−0.906 − 1.56i)17-s + (−0.344 + 0.596i)19-s − 0.927·20-s + 0.244·22-s + (−0.392 + 0.679i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 1.73i)T \)
13 \( 1 + T \)
good2 \( 1 + (0.190 - 0.330i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.11 - 1.93i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.73 + 6.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.88 - 3.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.35 + 7.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-0.736 + 1.27i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.736 - 1.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.73 - 6.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 + (5.35 + 9.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.35 - 9.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.11 - 1.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−9.987500996577917549745758689212, −9.010846694561937209699595201593, −8.005039148951372656891182649583, −7.24395070788793534298071109941, −6.80525128809619528921993459716, −5.77584626204260266496645621454, −4.20332010091474315354451546956, −3.29793360871651540590775749793, −2.58352746159938435446194787789, 0, 1.74653484422290677461957236649, 2.82323335618506260299832368688, 4.37097026767864617643545308797, 5.14878668267496541480930348675, 6.29066252686543234164784665600, 6.84514663199993461863276043130, 8.312429183163123294626687571934, 8.785139780967327474852124639337, 9.864856940100742380157567092400

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