Properties

Label 2-819-117.43-c1-0-40
Degree $2$
Conductor $819$
Sign $0.813 + 0.581i$
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  + (−0.836 + 0.483i)2-s + (−0.130 + 1.72i)3-s + (−0.533 + 0.923i)4-s + (−2.30 + 1.32i)5-s + (−0.725 − 1.50i)6-s i·7-s − 2.96i·8-s + (−2.96 − 0.450i)9-s + (1.28 − 2.22i)10-s + (−4.31 + 2.49i)11-s + (−1.52 − 1.04i)12-s + (3.52 − 0.770i)13-s + (0.483 + 0.836i)14-s + (−1.99 − 4.15i)15-s + (0.364 + 0.631i)16-s + (−0.434 − 0.751i)17-s + ⋯
L(s)  = 1  + (−0.591 + 0.341i)2-s + (−0.0752 + 0.997i)3-s + (−0.266 + 0.461i)4-s + (−1.03 + 0.594i)5-s + (−0.296 − 0.615i)6-s − 0.377i·7-s − 1.04i·8-s + (−0.988 − 0.150i)9-s + (0.406 − 0.703i)10-s + (−1.30 + 0.751i)11-s + (−0.440 − 0.300i)12-s + (0.976 − 0.213i)13-s + (0.129 + 0.223i)14-s + (−0.515 − 1.07i)15-s + (0.0910 + 0.157i)16-s + (−0.105 − 0.182i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.813 + 0.581i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.813 + 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128222 - 0.0411246i\)
\(L(\frac12)\) \(\approx\) \(0.128222 - 0.0411246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.130 - 1.72i)T \)
7 \( 1 + iT \)
13 \( 1 + (-3.52 + 0.770i)T \)
good2 \( 1 + (0.836 - 0.483i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (2.30 - 1.32i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.31 - 2.49i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.434 + 0.751i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.00 + 0.580i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.789T + 23T^{2} \)
29 \( 1 + (3.47 + 6.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.97 - 2.87i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.20 + 3.00i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.57iT - 41T^{2} \)
43 \( 1 - 7.28T + 43T^{2} \)
47 \( 1 + (-7.19 - 4.15i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.46T + 53T^{2} \)
59 \( 1 + (7.40 + 4.27i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 0.695iT - 67T^{2} \)
71 \( 1 + (1.85 - 1.07i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.80iT - 73T^{2} \)
79 \( 1 + (-1.13 + 1.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.01 + 4.62i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (11.7 + 6.77i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.8iT - 97T^{2} \)
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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

−10.12225632748759373980731645081, −9.292031365315676702133692285752, −8.388764657245422106169824774685, −7.68195782088518324546423859687, −7.08895406126148848402929832858, −5.71795240338052313012740254420, −4.49993087281527828612762403392, −3.78688537651356979626868400653, −2.92822202916628914246003402893, −0.099574905343187695423120199461, 1.06760336114759033541621337078, 2.39232454084327896425474491380, 3.77855820277907976424442812155, 5.33200220442550208938116401214, 5.70841019795506939718835913855, 7.12793821822797524465969286499, 8.060617030054762220261681993443, 8.565985486914945569853719905063, 9.128373580707331763689679742781, 10.64280992102888004373111533203

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