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Properties

Label 2-819-13.9-c1-0-16
Degree $2$
Conductor $819$
Sign $-0.500 - 0.866i$
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  + (−1.18 + 2.05i)2-s + (−1.82 − 3.16i)4-s + 4.02·5-s + (0.5 + 0.866i)7-s + 3.92·8-s + (−4.78 + 8.29i)10-s + (−1.63 + 2.83i)11-s + (0.910 − 3.48i)13-s − 2.37·14-s + (−1.01 + 1.75i)16-s + (−0.188 − 0.326i)17-s + (1.77 + 3.07i)19-s + (−7.35 − 12.7i)20-s + (−3.89 − 6.74i)22-s + (0.948 − 1.64i)23-s + ⋯
L(s)  = 1  + (−0.840 + 1.45i)2-s + (−0.912 − 1.58i)4-s + 1.80·5-s + (0.188 + 0.327i)7-s + 1.38·8-s + (−1.51 + 2.62i)10-s + (−0.493 + 0.854i)11-s + (0.252 − 0.967i)13-s − 0.635·14-s + (−0.253 + 0.439i)16-s + (−0.0457 − 0.0792i)17-s + (0.406 + 0.704i)19-s + (−1.64 − 2.84i)20-s + (−0.829 − 1.43i)22-s + (0.197 − 0.342i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.500 - 0.866i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.500 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.658457 + 1.14048i\)
\(L(\frac12)\) \(\approx\) \(0.658457 + 1.14048i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.910 + 3.48i)T \)
good2 \( 1 + (1.18 - 2.05i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 4.02T + 5T^{2} \)
11 \( 1 + (1.63 - 2.83i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.188 + 0.326i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.77 - 3.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.948 + 1.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.33 - 7.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.33T + 31T^{2} \)
37 \( 1 + (2.01 - 3.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.75 + 6.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.32 - 7.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + 7.75T + 53T^{2} \)
59 \( 1 + (1.05 + 1.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.87 + 6.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.79 - 4.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.99 - 3.45i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.50T + 73T^{2} \)
79 \( 1 + 1.16T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 + (-3.17 + 5.50i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.48 + 4.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.15087936226661543208439227213, −9.490804398129328529038178864742, −8.840985879281237640972369450716, −7.925946011293883048709779814100, −7.08385285579582429217256797578, −6.15662962853553494060667602084, −5.56552939388758805392276848707, −4.92311083969833359556021391684, −2.72683131341486359002059884524, −1.36865416409567737178647772933, 1.00130748461816562675267790374, 2.09646019832130152317869623911, 2.85667376116803352916420187894, 4.21333809033189821207273493965, 5.53462259184677514671361324627, 6.38532299927579978447469516991, 7.69257979224937476081230600337, 8.842407187402676572783485733823, 9.289916901551716137079350865674, 9.970993753003772444936239902574

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