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Properties

Label 2-819-91.76-c1-0-15
Degree $2$
Conductor $819$
Sign $0.131 - 0.991i$
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  + (−0.218 + 0.813i)2-s + (1.11 + 0.645i)4-s + (1.01 − 1.01i)5-s + (−2.62 + 0.330i)7-s + (−1.95 + 1.95i)8-s + (0.603 + 1.04i)10-s + (4.68 + 1.25i)11-s + (1.04 − 3.44i)13-s + (0.303 − 2.20i)14-s + (0.122 + 0.212i)16-s + (−1.49 + 2.58i)17-s + (1.62 + 6.06i)19-s + (1.78 − 0.478i)20-s + (−2.04 + 3.54i)22-s + (1.02 − 0.590i)23-s + ⋯
L(s)  = 1  + (−0.154 + 0.575i)2-s + (0.558 + 0.322i)4-s + (0.453 − 0.453i)5-s + (−0.992 + 0.124i)7-s + (−0.692 + 0.692i)8-s + (0.190 + 0.330i)10-s + (1.41 + 0.378i)11-s + (0.291 − 0.956i)13-s + (0.0811 − 0.590i)14-s + (0.0306 + 0.0531i)16-s + (−0.361 + 0.626i)17-s + (0.372 + 1.39i)19-s + (0.399 − 0.106i)20-s + (−0.435 + 0.755i)22-s + (0.213 − 0.123i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.131 - 0.991i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.131 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27609 + 1.11822i\)
\(L(\frac12)\) \(\approx\) \(1.27609 + 1.11822i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.330i)T \)
13 \( 1 + (-1.04 + 3.44i)T \)
good2 \( 1 + (0.218 - 0.813i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.01 + 1.01i)T - 5iT^{2} \)
11 \( 1 + (-4.68 - 1.25i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.49 - 2.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.62 - 6.06i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.02 + 0.590i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.77 - 4.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.79 + 4.79i)T - 31iT^{2} \)
37 \( 1 + (-5.40 - 1.44i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.71 + 1.26i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.90 + 1.67i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.12 - 4.12i)T + 47iT^{2} \)
53 \( 1 + 5.79T + 53T^{2} \)
59 \( 1 + (-10.7 + 2.88i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (7.95 + 4.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.508 - 1.89i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.19 + 0.855i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.353 + 0.353i)T + 73iT^{2} \)
79 \( 1 + 6.95T + 79T^{2} \)
83 \( 1 + (3.22 - 3.22i)T - 83iT^{2} \)
89 \( 1 + (-0.0636 + 0.237i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.43 - 9.08i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.22984508011771786210994715654, −9.481490672374415002636043146353, −8.675915963107737676084290504051, −7.88675235884149373305367148680, −6.79722430643707252190301028787, −6.21912951660392188726274533771, −5.49335754186761188358788339473, −3.93946701929928294821187226425, −2.99718319920465557156505246980, −1.51378651937622934617370322303, 0.958544159635060214388108982286, 2.40794845962334264835541315015, 3.23624645681482693575721596766, 4.41073705858259960972152831801, 5.98772692590617188866470976211, 6.68340634408248848697968295831, 6.94967837438992793584364177156, 8.759793325647372853382396362256, 9.436396745514677709963412119585, 9.966156819408802710558934358241

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