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Properties

Label 2-819-91.6-c1-0-28
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: $\chi_{819} (370, \cdot )$
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

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

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