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Properties

Label 2-819-91.76-c1-0-33
Degree $2$
Conductor $819$
Sign $0.762 + 0.647i$
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  + (−0.0578 + 0.215i)2-s + (1.68 + 0.975i)4-s + (1.08 − 1.08i)5-s + (−0.643 − 2.56i)7-s + (−0.624 + 0.624i)8-s + (0.171 + 0.297i)10-s + (−4.17 − 1.11i)11-s + (3.01 − 1.98i)13-s + (0.591 + 0.00954i)14-s + (1.85 + 3.20i)16-s + (3.78 − 6.55i)17-s + (−1.31 − 4.91i)19-s + (2.89 − 0.776i)20-s + (0.482 − 0.836i)22-s + (1.85 − 1.07i)23-s + ⋯
L(s)  = 1  + (−0.0408 + 0.152i)2-s + (0.844 + 0.487i)4-s + (0.486 − 0.486i)5-s + (−0.243 − 0.969i)7-s + (−0.220 + 0.220i)8-s + (0.0543 + 0.0941i)10-s + (−1.25 − 0.337i)11-s + (0.835 − 0.550i)13-s + (0.157 + 0.00255i)14-s + (0.462 + 0.801i)16-s + (0.917 − 1.58i)17-s + (−0.302 − 1.12i)19-s + (0.647 − 0.173i)20-s + (0.102 − 0.178i)22-s + (0.387 − 0.223i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.762 + 0.647i$
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: \((2,\ 819,\ (\ :1/2),\ 0.762 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74246 - 0.640030i\)
\(L(\frac12)\) \(\approx\) \(1.74246 - 0.640030i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.643 + 2.56i)T \)
13 \( 1 + (-3.01 + 1.98i)T \)
good2 \( 1 + (0.0578 - 0.215i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.08 + 1.08i)T - 5iT^{2} \)
11 \( 1 + (4.17 + 1.11i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.78 + 6.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.31 + 4.91i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.85 + 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.66 - 4.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.09 - 1.09i)T - 31iT^{2} \)
37 \( 1 + (2.82 + 0.757i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.37 - 0.369i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.58 + 3.80i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.26 - 5.26i)T + 47iT^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 + (8.73 - 2.34i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-10.1 - 5.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.28 - 4.78i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-5.25 + 1.40i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (8.04 + 8.04i)T + 73iT^{2} \)
79 \( 1 - 9.06T + 79T^{2} \)
83 \( 1 + (-1.13 + 1.13i)T - 83iT^{2} \)
89 \( 1 + (0.217 - 0.811i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.35 - 16.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.41169184133170615977560576694, −9.212710796873662274474123199729, −8.360884950496829641710884673451, −7.41805944567292373694401189866, −6.92196147999417100940392914517, −5.69678936215591390870372698279, −4.95984626113388549268250373963, −3.42912850510503464313240258984, −2.66488027253860708908227600486, −0.948371498270707875238197466298, 1.73254242400188120531459523988, 2.52781086260790085479320265080, 3.67182074112351861135683083775, 5.37465242653350047237776822167, 5.99896461183538241779733343778, 6.58997232296710506059912431829, 7.84112054166172380527255980146, 8.569922775679763118331165727214, 9.900953829904244309839710212420, 10.23423285201542127760203859191

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