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Properties

Label 2-819-91.54-c1-0-13
Degree $2$
Conductor $819$
Sign $0.278 - 0.960i$
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  + (0.240 − 0.240i)2-s + 1.88i·4-s + (−2.06 + 0.552i)5-s + (1.35 − 2.27i)7-s + (0.935 + 0.935i)8-s + (−0.363 + 0.629i)10-s + (−0.208 + 0.0558i)11-s + (3.58 + 0.408i)13-s + (−0.219 − 0.873i)14-s − 3.31·16-s + 5.29·17-s + (−1.66 + 6.21i)19-s + (−1.04 − 3.88i)20-s + (−0.0367 + 0.0636i)22-s + 6.44i·23-s + ⋯
L(s)  = 1  + (0.170 − 0.170i)2-s + 0.942i·4-s + (−0.922 + 0.247i)5-s + (0.513 − 0.858i)7-s + (0.330 + 0.330i)8-s + (−0.115 + 0.199i)10-s + (−0.0628 + 0.0168i)11-s + (0.993 + 0.113i)13-s + (−0.0586 − 0.233i)14-s − 0.829·16-s + 1.28·17-s + (−0.381 + 1.42i)19-s + (−0.232 − 0.869i)20-s + (−0.00783 + 0.0135i)22-s + 1.34i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.278 - 0.960i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.278 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15188 + 0.865658i\)
\(L(\frac12)\) \(\approx\) \(1.15188 + 0.865658i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 + 2.27i)T \)
13 \( 1 + (-3.58 - 0.408i)T \)
good2 \( 1 + (-0.240 + 0.240i)T - 2iT^{2} \)
5 \( 1 + (2.06 - 0.552i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.208 - 0.0558i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 + (1.66 - 6.21i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 6.44iT - 23T^{2} \)
29 \( 1 + (1.21 + 2.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.681 - 2.54i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.73 - 2.73i)T + 37iT^{2} \)
41 \( 1 + (2.28 - 8.53i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.95 + 2.85i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.31 - 12.3i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.97 + 3.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.64 + 8.64i)T - 59iT^{2} \)
61 \( 1 + (2.44 - 1.40i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.436 - 1.62i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.653 + 2.44i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.0329 - 0.00884i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.28 + 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.72 - 7.72i)T + 83iT^{2} \)
89 \( 1 + (-5.91 + 5.91i)T - 89iT^{2} \)
97 \( 1 + (-7.90 + 2.11i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.60804124899890555538560191290, −9.645466703845587238437757583897, −8.280438356825473751711861767567, −7.903145858182203905610731859762, −7.31397829747812253809569599870, −6.09355960102368310754215887351, −4.76614787546593972810072514480, −3.65126318993842558037889112704, −3.48061898178287654268570990030, −1.52462366513074212655945961261, 0.73961182047643664827847958007, 2.27605590782145899307280950879, 3.78419444823069073761866115681, 4.81276931823650545098169958177, 5.55812124984686861788640149803, 6.45893625624658659274617040124, 7.49899437467175903923232142810, 8.509412403493494592222286319474, 8.995666337691192998810474940470, 10.17944485724957070269115713409

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