Properties

Label 2-819-13.9-c1-0-9
Degree $2$
Conductor $819$
Sign $-0.0128 - 0.999i$
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  + (0.190 − 0.330i)2-s + (0.927 + 1.60i)4-s − 0.381·5-s + (−0.5 − 0.866i)7-s + 1.47·8-s + (−0.0729 + 0.126i)10-s + (−2.42 + 4.20i)11-s + (−2.5 + 2.59i)13-s − 0.381·14-s + (−1.57 + 2.72i)16-s + (3.73 + 6.47i)17-s + (−2.42 − 4.20i)19-s + (−0.354 − 0.613i)20-s + (0.927 + 1.60i)22-s + (2.23 − 3.87i)23-s + ⋯
L(s)  = 1  + (0.135 − 0.233i)2-s + (0.463 + 0.802i)4-s − 0.170·5-s + (−0.188 − 0.327i)7-s + 0.520·8-s + (−0.0230 + 0.0399i)10-s + (−0.731 + 1.26i)11-s + (−0.693 + 0.720i)13-s − 0.102·14-s + (−0.393 + 0.681i)16-s + (0.906 + 1.56i)17-s + (−0.556 − 0.964i)19-s + (−0.0791 − 0.137i)20-s + (0.197 + 0.342i)22-s + (0.466 − 0.807i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.0128 - 0.999i$
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: \((2,\ 819,\ (\ :1/2),\ -0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.998608 + 1.01149i\)
\(L(\frac12)\) \(\approx\) \(0.998608 + 1.01149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 2.59i)T \)
good2 \( 1 + (-0.190 + 0.330i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 0.381T + 5T^{2} \)
11 \( 1 + (2.42 - 4.20i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.73 - 6.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.42 + 4.20i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.23 + 3.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.04 - 3.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.70T + 31T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.61 + 4.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.78 - 6.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.23T + 47T^{2} \)
53 \( 1 + 8.23T + 53T^{2} \)
59 \( 1 + (-1.11 - 1.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.354 - 0.613i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.09 - 7.08i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 + (-8.04 + 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.07 + 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.45110438994788950830886850223, −9.824694925727353730363959051922, −8.609432880264323150680285941877, −7.78293430802580175255287716455, −7.12577117877480719011546405196, −6.30265891331899111803496247732, −4.76644028616231424088351591976, −4.12717149339415676035531642755, −2.88839007343481812204518329279, −1.89232333274381743998561965559, 0.64321717806939043957696105151, 2.39181384031773604804189766146, 3.38013031077345341789765706428, 4.99795370004493506715759759109, 5.60194763657191085290964024000, 6.33743520295249215975961340096, 7.59407467048077706020535351884, 8.012866776448582930040988234484, 9.411136417832784912572796776136, 9.995202407898850301355382269874

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