Properties

Label 2-819-91.59-c1-0-44
Degree $2$
Conductor $819$
Sign $-0.607 - 0.794i$
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  + (−0.430 − 0.430i)2-s − 1.62i·4-s + (−1.97 − 0.529i)5-s + (−1.23 − 2.34i)7-s + (−1.56 + 1.56i)8-s + (0.623 + 1.08i)10-s + (0.0981 + 0.0263i)11-s + (3.09 − 1.85i)13-s + (−0.476 + 1.53i)14-s − 1.91·16-s − 3.63·17-s + (0.374 + 1.39i)19-s + (−0.863 + 3.22i)20-s + (−0.0309 − 0.0536i)22-s − 4.80i·23-s + ⋯
L(s)  = 1  + (−0.304 − 0.304i)2-s − 0.814i·4-s + (−0.884 − 0.236i)5-s + (−0.466 − 0.884i)7-s + (−0.552 + 0.552i)8-s + (0.197 + 0.341i)10-s + (0.0295 + 0.00792i)11-s + (0.858 − 0.513i)13-s + (−0.127 + 0.411i)14-s − 0.477·16-s − 0.880·17-s + (0.0858 + 0.320i)19-s + (−0.192 + 0.720i)20-s + (−0.00659 − 0.0114i)22-s − 1.00i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.607 - 0.794i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.607 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109033 + 0.220718i\)
\(L(\frac12)\) \(\approx\) \(0.109033 + 0.220718i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 + 2.34i)T \)
13 \( 1 + (-3.09 + 1.85i)T \)
good2 \( 1 + (0.430 + 0.430i)T + 2iT^{2} \)
5 \( 1 + (1.97 + 0.529i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.0981 - 0.0263i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 3.63T + 17T^{2} \)
19 \( 1 + (-0.374 - 1.39i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 4.80iT - 23T^{2} \)
29 \( 1 + (3.79 - 6.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.73 - 6.47i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.15 - 2.15i)T - 37iT^{2} \)
41 \( 1 + (-0.872 - 3.25i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.21 - 4.16i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.529 + 1.97i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.27 + 9.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.58 - 1.58i)T + 59iT^{2} \)
61 \( 1 + (5.15 + 2.97i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.62 - 6.05i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.0733 + 0.273i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.17 + 1.11i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.01 + 8.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.74 - 3.74i)T - 83iT^{2} \)
89 \( 1 + (5.75 + 5.75i)T + 89iT^{2} \)
97 \( 1 + (-16.5 - 4.43i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.874165873072624900786071217774, −8.788071966614846270953738358113, −8.231728775667153484918822527185, −7.00721615819453959758687216025, −6.32831500703574095043019451977, −5.12157572759340450754252517035, −4.14959649350844672937845432683, −3.12166800717007941942930429241, −1.42244248349205228145850014395, −0.13794142694934381037344443246, 2.35073981703580058814048935674, 3.54202492838186556472997997323, 4.18505357046823675493159631264, 5.74239288239625317295257360131, 6.62102969849940493245900667752, 7.45814680528763184865529377609, 8.216844391277066488059963449135, 9.012200176411466930700772974304, 9.565126079535341834433237713917, 11.02016590665186022654135641440

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