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Properties

Label 2-819-91.6-c1-0-32
Degree $2$
Conductor $819$
Sign $-0.536 + 0.843i$
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  + (−0.385 − 1.43i)2-s + (−0.189 + 0.109i)4-s + (1.07 + 1.07i)5-s + (1.81 − 1.92i)7-s + (−1.87 − 1.87i)8-s + (1.12 − 1.95i)10-s + (−1.68 + 0.451i)11-s + (3.51 − 0.818i)13-s + (−3.46 − 1.87i)14-s + (−2.19 + 3.80i)16-s + (−1.43 − 2.48i)17-s + (0.389 − 1.45i)19-s + (−0.319 − 0.0856i)20-s + (1.29 + 2.25i)22-s + (3.21 + 1.85i)23-s + ⋯
L(s)  = 1  + (−0.272 − 1.01i)2-s + (−0.0945 + 0.0545i)4-s + (0.479 + 0.479i)5-s + (0.686 − 0.726i)7-s + (−0.663 − 0.663i)8-s + (0.357 − 0.618i)10-s + (−0.508 + 0.136i)11-s + (0.973 − 0.227i)13-s + (−0.926 − 0.500i)14-s + (−0.548 + 0.950i)16-s + (−0.348 − 0.602i)17-s + (0.0893 − 0.333i)19-s + (−0.0714 − 0.0191i)20-s + (0.277 + 0.479i)22-s + (0.669 + 0.386i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.536 + 0.843i$
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: \((2,\ 819,\ (\ :1/2),\ -0.536 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771223 - 1.40491i\)
\(L(\frac12)\) \(\approx\) \(0.771223 - 1.40491i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 + 1.92i)T \)
13 \( 1 + (-3.51 + 0.818i)T \)
good2 \( 1 + (0.385 + 1.43i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-1.07 - 1.07i)T + 5iT^{2} \)
11 \( 1 + (1.68 - 0.451i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.43 + 2.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.389 + 1.45i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.21 - 1.85i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.65 + 2.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.32 - 1.32i)T + 31iT^{2} \)
37 \( 1 + (5.83 - 1.56i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.10 + 0.830i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.29 + 1.89i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.86 + 5.86i)T - 47iT^{2} \)
53 \( 1 - 1.37T + 53T^{2} \)
59 \( 1 + (0.967 + 0.259i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.0305 - 0.0176i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.26 + 4.70i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (11.4 + 3.05i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (11.3 - 11.3i)T - 73iT^{2} \)
79 \( 1 + 3.53T + 79T^{2} \)
83 \( 1 + (-10.8 - 10.8i)T + 83iT^{2} \)
89 \( 1 + (2.02 + 7.54i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.46 - 9.21i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.39763112233726223407967657551, −9.328194259148132182422935148027, −8.470042782005832200976088377362, −7.32713101577787196939115110419, −6.55904037394231092745866723279, −5.48531135692218094173653880052, −4.25766584073616831913700790219, −3.12463785697522483223375626011, −2.15521416967418962266728876240, −0.915717424960874256575551983240, 1.64856008859521544798042410280, 2.93573855271387956223466673727, 4.55493438920945959523216893104, 5.60766480926608158434435249769, 6.01530196383273633428725579285, 7.13276039381927060060709264727, 8.045659054418417641302018647509, 8.763405224680277902777885602866, 9.144052974696022012214213783326, 10.56630622189976617754531844575

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