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Properties

Label 2-819-91.33-c1-0-11
Degree $2$
Conductor $819$
Sign $-0.987 - 0.154i$
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  + (−0.306 + 1.14i)2-s + (0.518 + 0.299i)4-s + (0.424 + 1.58i)5-s + (1.65 + 2.06i)7-s + (−2.17 + 2.17i)8-s − 1.94·10-s + (−2.06 + 2.06i)11-s + (−2.68 − 2.40i)13-s + (−2.86 + 1.26i)14-s + (−1.22 − 2.11i)16-s + (0.405 − 0.702i)17-s + (−4.56 + 4.56i)19-s + (−0.253 + 0.947i)20-s + (−1.72 − 2.99i)22-s + (1.58 − 0.917i)23-s + ⋯
L(s)  = 1  + (−0.216 + 0.808i)2-s + (0.259 + 0.149i)4-s + (0.189 + 0.708i)5-s + (0.626 + 0.779i)7-s + (−0.769 + 0.769i)8-s − 0.613·10-s + (−0.621 + 0.621i)11-s + (−0.743 − 0.668i)13-s + (−0.765 + 0.338i)14-s + (−0.305 − 0.529i)16-s + (0.0983 − 0.170i)17-s + (−1.04 + 1.04i)19-s + (−0.0567 + 0.211i)20-s + (−0.368 − 0.637i)22-s + (0.331 − 0.191i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.987 - 0.154i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.987 - 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105215 + 1.35148i\)
\(L(\frac12)\) \(\approx\) \(0.105215 + 1.35148i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 - 2.06i)T \)
13 \( 1 + (2.68 + 2.40i)T \)
good2 \( 1 + (0.306 - 1.14i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-0.424 - 1.58i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.06 - 2.06i)T - 11iT^{2} \)
17 \( 1 + (-0.405 + 0.702i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.56 - 4.56i)T - 19iT^{2} \)
23 \( 1 + (-1.58 + 0.917i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.42 + 4.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.78 - 1.01i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.95 - 1.06i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.392 + 1.46i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.14 - 1.23i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.21 + 0.325i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.12 - 1.95i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.46 - 1.73i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 0.516iT - 61T^{2} \)
67 \( 1 + (-9.11 - 9.11i)T + 67iT^{2} \)
71 \( 1 + (-2.21 + 8.27i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.64 - 9.87i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.17 - 3.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.44 - 7.44i)T - 83iT^{2} \)
89 \( 1 + (-2.18 + 8.16i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (16.5 + 4.44i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.52663610708357871469275629411, −9.834281947858664708320339776444, −8.573714148230000259849010977426, −8.026979411367264252695762477141, −7.25974170225647691247779618343, −6.35765931824857563790495336504, −5.60242602561082884958431193265, −4.62363517729918777040057659382, −2.86893158371137442444315990586, −2.23809940404405971269318066707, 0.68462787792315583693084381581, 1.88312842125148764460009441153, 3.01583253815965327667498026057, 4.38299028616576612764432564153, 5.15666087362088368836538382800, 6.42164816492105340097058301396, 7.24272997709986888432791563584, 8.359675495946483988183767676789, 9.119722240664861138459628085735, 9.982428652897630742004482765040

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