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Properties

Label 2-819-13.4-c1-0-26
Degree $2$
Conductor $819$
Sign $0.607 + 0.794i$
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  + (1.20 + 0.692i)2-s + (−0.0395 − 0.0685i)4-s − 0.518i·5-s + (−0.866 + 0.5i)7-s − 2.88i·8-s + (0.359 − 0.622i)10-s + (−1.40 − 0.812i)11-s + (1.42 − 3.31i)13-s − 1.38·14-s + (1.91 − 3.32i)16-s + (−0.974 − 1.68i)17-s + (2.15 − 1.24i)19-s + (−0.0355 + 0.0205i)20-s + (−1.12 − 1.94i)22-s + (4.57 − 7.91i)23-s + ⋯
L(s)  = 1  + (0.848 + 0.490i)2-s + (−0.0197 − 0.0342i)4-s − 0.232i·5-s + (−0.327 + 0.188i)7-s − 1.01i·8-s + (0.113 − 0.196i)10-s + (−0.424 − 0.244i)11-s + (0.395 − 0.918i)13-s − 0.370·14-s + (0.479 − 0.830i)16-s + (−0.236 − 0.409i)17-s + (0.494 − 0.285i)19-s + (−0.00795 + 0.00459i)20-s + (−0.239 − 0.415i)22-s + (0.952 − 1.65i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.607 + 0.794i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.607 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78905 - 0.883617i\)
\(L(\frac12)\) \(\approx\) \(1.78905 - 0.883617i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-1.42 + 3.31i)T \)
good2 \( 1 + (-1.20 - 0.692i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 0.518iT - 5T^{2} \)
11 \( 1 + (1.40 + 0.812i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.974 + 1.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.15 + 1.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.57 + 7.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.61 - 4.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.79iT - 31T^{2} \)
37 \( 1 + (8.85 + 5.11i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.64 + 2.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.51iT - 47T^{2} \)
53 \( 1 - 8.89T + 53T^{2} \)
59 \( 1 + (-5.37 + 3.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.73 - 11.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.25 + 4.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.50 + 2.59i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 - 0.982T + 79T^{2} \)
83 \( 1 - 8.91iT - 83T^{2} \)
89 \( 1 + (-10.4 - 6.00i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.82 + 2.21i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.34705654614256033128464151004, −9.059745585259824050974171078860, −8.553518573578782746841388738273, −7.13500689061738112624281446924, −6.63763149531211340517591122798, −5.26381896807273538264309000050, −5.20894598289341981159785476210, −3.77491352402900870702882440126, −2.82110013419334690318718603873, −0.77786669666716534585958144505, 1.81566407701361667811407595255, 3.09062465876099551027333410802, 3.84837103130054454767997464209, 4.84555232233497446368040545124, 5.74368907671042518059640631009, 6.85053821870840278854413035452, 7.73328683974676431841743729805, 8.728884562082296541679012528866, 9.582674264549238918668449350845, 10.56502958129045675527879286207

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