Properties

Label 2-819-91.41-c1-0-6
Degree $2$
Conductor $819$
Sign $0.998 + 0.0587i$
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  + (−1.34 − 0.359i)2-s + (−0.0625 − 0.0361i)4-s + (−2.52 − 2.52i)5-s + (−0.324 + 2.62i)7-s + (2.03 + 2.03i)8-s + (2.48 + 4.29i)10-s + (−0.0529 + 0.197i)11-s + (−2.07 − 2.94i)13-s + (1.37 − 3.40i)14-s + (−1.92 − 3.33i)16-s + (−1.13 + 1.97i)17-s + (−1.50 + 0.402i)19-s + (0.0668 + 0.249i)20-s + (0.142 − 0.246i)22-s + (−2.59 + 1.49i)23-s + ⋯
L(s)  = 1  + (−0.948 − 0.254i)2-s + (−0.0312 − 0.0180i)4-s + (−1.13 − 1.13i)5-s + (−0.122 + 0.992i)7-s + (0.719 + 0.719i)8-s + (0.785 + 1.35i)10-s + (−0.0159 + 0.0596i)11-s + (−0.575 − 0.817i)13-s + (0.368 − 0.910i)14-s + (−0.481 − 0.833i)16-s + (−0.275 + 0.477i)17-s + (−0.344 + 0.0923i)19-s + (0.0149 + 0.0558i)20-s + (0.0302 − 0.0524i)22-s + (−0.541 + 0.312i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.998 + 0.0587i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (496, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.998 + 0.0587i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.499060 - 0.0146823i\)
\(L(\frac12)\) \(\approx\) \(0.499060 - 0.0146823i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.324 - 2.62i)T \)
13 \( 1 + (2.07 + 2.94i)T \)
good2 \( 1 + (1.34 + 0.359i)T + (1.73 + i)T^{2} \)
5 \( 1 + (2.52 + 2.52i)T + 5iT^{2} \)
11 \( 1 + (0.0529 - 0.197i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.13 - 1.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.50 - 0.402i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.59 - 1.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.75 - 8.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.75 - 2.75i)T + 31iT^{2} \)
37 \( 1 + (-1.17 + 4.39i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.36 + 5.07i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.76 - 1.01i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.42 + 9.42i)T - 47iT^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + (-0.510 - 1.90i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.850 + 0.491i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-15.1 - 4.06i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.00355 + 0.0132i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.24 + 2.24i)T - 73iT^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (3.37 + 3.37i)T + 83iT^{2} \)
89 \( 1 + (11.8 + 3.17i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.85 - 2.10i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.14893757713801805171773826859, −9.123843292587695628184299355227, −8.594254160009567439198285150827, −8.122470302598237045886622994524, −7.15723726811188334205678993475, −5.56400575503804786447748022501, −4.94303455251742240588496758094, −3.85370594642392075678537557804, −2.28924514506786847267488665132, −0.790415361360052110827845855106, 0.54048452610291996208673308805, 2.64136013871982916460838236383, 4.04875778310325627897797087326, 4.38129054374549807671357630655, 6.47793328387666131979925977365, 7.00200846780127395128102213558, 7.75673187566486253538295331512, 8.280984675261566800586246677298, 9.533133032046761033622990348372, 10.10685545353620582650521458826

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