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Properties

Label 2-819-117.2-c1-0-39
Degree $2$
Conductor $819$
Sign $0.290 - 0.956i$
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  + (0.671 + 0.671i)2-s + (1.42 + 0.980i)3-s − 1.09i·4-s + (−0.751 + 2.80i)5-s + (0.300 + 1.61i)6-s + (0.258 − 0.965i)7-s + (2.07 − 2.07i)8-s + (1.07 + 2.79i)9-s + (−2.38 + 1.37i)10-s + (2.96 − 2.96i)11-s + (1.07 − 1.56i)12-s + (1.14 + 3.41i)13-s + (0.821 − 0.474i)14-s + (−3.82 + 3.26i)15-s + 0.593·16-s + (0.0848 − 0.146i)17-s + ⋯
L(s)  = 1  + (0.474 + 0.474i)2-s + (0.824 + 0.565i)3-s − 0.549i·4-s + (−0.336 + 1.25i)5-s + (0.122 + 0.659i)6-s + (0.0978 − 0.365i)7-s + (0.735 − 0.735i)8-s + (0.359 + 0.933i)9-s + (−0.754 + 0.435i)10-s + (0.895 − 0.895i)11-s + (0.310 − 0.453i)12-s + (0.317 + 0.948i)13-s + (0.219 − 0.126i)14-s + (−0.986 + 0.843i)15-s + 0.148·16-s + (0.0205 − 0.0356i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.290 - 0.956i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.290 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17014 + 1.60947i\)
\(L(\frac12)\) \(\approx\) \(2.17014 + 1.60947i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.42 - 0.980i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-1.14 - 3.41i)T \)
good2 \( 1 + (-0.671 - 0.671i)T + 2iT^{2} \)
5 \( 1 + (0.751 - 2.80i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.96 + 2.96i)T - 11iT^{2} \)
17 \( 1 + (-0.0848 + 0.146i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.798 - 2.98i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.02 - 1.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.29iT - 29T^{2} \)
31 \( 1 + (-2.53 - 0.680i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.20 - 8.23i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.49 - 0.668i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (7.99 - 4.61i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.97 + 7.38i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 6.78iT - 53T^{2} \)
59 \( 1 + (-4.35 + 4.35i)T - 59iT^{2} \)
61 \( 1 + (-3.75 - 6.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.40 + 12.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-14.0 + 3.76i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.87 + 4.87i)T + 73iT^{2} \)
79 \( 1 + (-6.13 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.92 - 2.12i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (4.91 + 1.31i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.86 + 2.37i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.20222823045318440354971355484, −9.799028269148064296210230326415, −8.657209774443314241896522950202, −7.75573618675306025808902967424, −6.75306463673149633919453518245, −6.28363191108612469777633004502, −4.93634314812606412638026726300, −3.89061390780735005471165844808, −3.33890344572423603720414805912, −1.72353473682473406598908100804, 1.25768994226497114063941048019, 2.43921854639781944091958324127, 3.59108182017494278888512264595, 4.39919370182129394924957803196, 5.33184168904910770864106971589, 6.82619775399684186170199447515, 7.65497865877199338772018788620, 8.553756419838691977102671240168, 8.843656031127939024133843992679, 9.908013858883107879749779967785

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