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Properties

Label 2-819-91.80-c1-0-25
Degree $2$
Conductor $819$
Sign $0.164 + 0.986i$
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  + (−0.255 − 0.954i)2-s + (0.886 − 0.511i)4-s + (0.244 − 0.912i)5-s + (2.46 − 0.953i)7-s + (−2.11 − 2.11i)8-s − 0.933·10-s + (4.64 + 4.64i)11-s + (3.43 + 1.08i)13-s + (−1.54 − 2.11i)14-s + (−0.452 + 0.783i)16-s + (−1.86 − 3.23i)17-s + (0.889 + 0.889i)19-s + (−0.250 − 0.933i)20-s + (3.24 − 5.62i)22-s + (−0.202 − 0.116i)23-s + ⋯
L(s)  = 1  + (−0.180 − 0.674i)2-s + (0.443 − 0.255i)4-s + (0.109 − 0.407i)5-s + (0.932 − 0.360i)7-s + (−0.746 − 0.746i)8-s − 0.295·10-s + (1.40 + 1.40i)11-s + (0.953 + 0.299i)13-s + (−0.411 − 0.564i)14-s + (−0.113 + 0.195i)16-s + (−0.452 − 0.784i)17-s + (0.204 + 0.204i)19-s + (−0.0559 − 0.208i)20-s + (0.692 − 1.19i)22-s + (−0.0421 − 0.0243i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.164 + 0.986i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.164 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50206 - 1.27242i\)
\(L(\frac12)\) \(\approx\) \(1.50206 - 1.27242i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.46 + 0.953i)T \)
13 \( 1 + (-3.43 - 1.08i)T \)
good2 \( 1 + (0.255 + 0.954i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.244 + 0.912i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.64 - 4.64i)T + 11iT^{2} \)
17 \( 1 + (1.86 + 3.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.889 - 0.889i)T + 19iT^{2} \)
23 \( 1 + (0.202 + 0.116i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.32 - 4.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.35 - 1.96i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.85 - 1.30i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.49 + 9.29i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (10.4 + 6.00i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.10 - 0.563i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.04 - 3.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.20 - 1.93i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 2.45iT - 61T^{2} \)
67 \( 1 + (7.19 - 7.19i)T - 67iT^{2} \)
71 \( 1 + (0.433 + 1.61i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (2.10 + 7.85i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.942 + 1.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.95 + 9.95i)T + 83iT^{2} \)
89 \( 1 + (1.05 + 3.94i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.41 - 0.647i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.22325590012893332222375081442, −9.123602323285815431530742089867, −8.806366634571490109704382128007, −7.15155884234318896081153085879, −6.89521908661316362408221747848, −5.52306087632027993886198149242, −4.50053296364334436281768087338, −3.53819954612933041292050770108, −1.89794814391572134889396381748, −1.29889929687456177872333377634, 1.52836360104445761145938577418, 2.96487876703632804586411815394, 3.97205744808142396039069625440, 5.47457013659650278622343171413, 6.25864355054476003785475423230, 6.77093303727133738072667077558, 8.114288763153700273298306436067, 8.433409022500844569654639229081, 9.226985575059941485823368483651, 10.68571355319155877299416166077

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