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Properties

Label 2-819-91.59-c1-0-18
Degree $2$
Conductor $819$
Sign $-0.671 - 0.740i$
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  + (1.74 + 1.74i)2-s + 4.09i·4-s + (2.78 + 0.746i)5-s + (−2.58 − 0.573i)7-s + (−3.65 + 3.65i)8-s + (3.55 + 6.16i)10-s + (1.35 + 0.362i)11-s + (3.49 + 0.872i)13-s + (−3.50 − 5.51i)14-s − 4.58·16-s − 7.55·17-s + (1.78 + 6.67i)19-s + (−3.05 + 11.4i)20-s + (1.72 + 2.99i)22-s + 2.27i·23-s + ⋯
L(s)  = 1  + (1.23 + 1.23i)2-s + 2.04i·4-s + (1.24 + 0.333i)5-s + (−0.976 − 0.216i)7-s + (−1.29 + 1.29i)8-s + (1.12 + 1.94i)10-s + (0.408 + 0.109i)11-s + (0.970 + 0.242i)13-s + (−0.937 − 1.47i)14-s − 1.14·16-s − 1.83·17-s + (0.410 + 1.53i)19-s + (−0.683 + 2.55i)20-s + (0.368 + 0.638i)22-s + 0.475i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.671 - 0.740i$
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: \((2,\ 819,\ (\ :1/2),\ -0.671 - 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33583 + 3.01566i\)
\(L(\frac12)\) \(\approx\) \(1.33583 + 3.01566i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 + 0.573i)T \)
13 \( 1 + (-3.49 - 0.872i)T \)
good2 \( 1 + (-1.74 - 1.74i)T + 2iT^{2} \)
5 \( 1 + (-2.78 - 0.746i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.35 - 0.362i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 7.55T + 17T^{2} \)
19 \( 1 + (-1.78 - 6.67i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 2.27iT - 23T^{2} \)
29 \( 1 + (-3.75 + 6.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.39 + 5.19i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.61 + 1.61i)T - 37iT^{2} \)
41 \( 1 + (2.38 + 8.90i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.04 + 1.75i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.223 + 0.833i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.886 + 1.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.80 + 3.80i)T + 59iT^{2} \)
61 \( 1 + (3.62 + 2.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.44 - 5.38i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.75 + 6.54i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.10 + 2.17i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.411 - 0.713i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.15 - 2.15i)T - 83iT^{2} \)
89 \( 1 + (1.81 + 1.81i)T + 89iT^{2} \)
97 \( 1 + (-8.79 - 2.35i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.45298472515511260362748577081, −9.583248787576372154810336769704, −8.751751733313764874691104178086, −7.59853551222814678741737596527, −6.59619128093508579081126953936, −6.20993281460211487068165964873, −5.62848911446740916410225377125, −4.26000103551163001370466772484, −3.55901055420033611200893613463, −2.17788629671585849515600907923, 1.21575900591733046452850672220, 2.45199092888050928710256249437, 3.20215343886378504233072883973, 4.45423598501681711195838922834, 5.23010136046497037572508102461, 6.27109556849349677832915489366, 6.64844732121125886476295026273, 8.857544957361931182854102403715, 9.211436186282250246037728373145, 10.19810103022329336331192249758

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