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Properties

Label 2-819-91.76-c1-0-1
Degree $2$
Conductor $819$
Sign $0.997 - 0.0684i$
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  + (0.706 − 2.63i)2-s + (−4.72 − 2.72i)4-s + (−2.18 + 2.18i)5-s + (−1.85 − 1.88i)7-s + (−6.66 + 6.66i)8-s + (4.21 + 7.30i)10-s + (0.456 + 0.122i)11-s + (2.45 − 2.64i)13-s + (−6.28 + 3.56i)14-s + (7.42 + 12.8i)16-s + (−1.14 + 1.97i)17-s + (1.51 + 5.66i)19-s + (16.2 − 4.36i)20-s + (0.645 − 1.11i)22-s + (−0.481 + 0.278i)23-s + ⋯
L(s)  = 1  + (0.499 − 1.86i)2-s + (−2.36 − 1.36i)4-s + (−0.976 + 0.976i)5-s + (−0.702 − 0.712i)7-s + (−2.35 + 2.35i)8-s + (1.33 + 2.30i)10-s + (0.137 + 0.0368i)11-s + (0.680 − 0.732i)13-s + (−1.67 + 0.953i)14-s + (1.85 + 3.21i)16-s + (−0.276 + 0.479i)17-s + (0.348 + 1.30i)19-s + (3.63 − 0.975i)20-s + (0.137 − 0.238i)22-s + (−0.100 + 0.0580i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.997 - 0.0684i$
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: \((2,\ 819,\ (\ :1/2),\ 0.997 - 0.0684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461421 + 0.0158181i\)
\(L(\frac12)\) \(\approx\) \(0.461421 + 0.0158181i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.85 + 1.88i)T \)
13 \( 1 + (-2.45 + 2.64i)T \)
good2 \( 1 + (-0.706 + 2.63i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (2.18 - 2.18i)T - 5iT^{2} \)
11 \( 1 + (-0.456 - 0.122i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.14 - 1.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.51 - 5.66i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.481 - 0.278i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.64 - 6.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.74 - 2.74i)T - 31iT^{2} \)
37 \( 1 + (6.41 + 1.71i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.49 + 0.400i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.08 - 2.93i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.55 + 6.55i)T + 47iT^{2} \)
53 \( 1 + 4.17T + 53T^{2} \)
59 \( 1 + (14.2 - 3.82i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.553 + 0.319i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.17 - 8.10i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.13 + 0.572i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.43 - 2.43i)T + 73iT^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (1.80 - 1.80i)T - 83iT^{2} \)
89 \( 1 + (0.363 - 1.35i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.25 - 12.1i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.64049164190195683322702881975, −9.930619539965601779996422711072, −8.839836869745397681000973559552, −7.87170861233449977279896955108, −6.66613798726365782835472265407, −5.55442646749618035750331997369, −4.22916261490815100285606986388, −3.48021815850621178189435291894, −3.09859766910326529817596680077, −1.41401372022741754829464967856, 0.21018403142446993206441486195, 3.25775983117767182137804598284, 4.33767875784717680995239863902, 4.89170631360783959387152377099, 5.97664354615917460396275574866, 6.68016705007157287208673638338, 7.57121427327909831162864366068, 8.414544234843967468308574955227, 9.012591189118723547017273103671, 9.517902609273109275343823512505

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