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Properties

Label 2-819-91.20-c1-0-39
Degree $2$
Conductor $819$
Sign $0.616 + 0.787i$
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  + (2.16 − 0.581i)2-s + (2.63 − 1.52i)4-s + (1.87 − 1.87i)5-s + (2.25 − 1.38i)7-s + (1.65 − 1.65i)8-s + (2.98 − 5.16i)10-s + (1.20 + 4.51i)11-s + (−1.52 + 3.26i)13-s + (4.07 − 4.31i)14-s + (−0.417 + 0.723i)16-s + (−2.18 − 3.78i)17-s + (−0.194 − 0.0521i)19-s + (2.09 − 7.80i)20-s + (5.24 + 9.08i)22-s + (−7.20 − 4.15i)23-s + ⋯
L(s)  = 1  + (1.53 − 0.410i)2-s + (1.31 − 0.760i)4-s + (0.839 − 0.839i)5-s + (0.851 − 0.524i)7-s + (0.584 − 0.584i)8-s + (0.942 − 1.63i)10-s + (0.364 + 1.36i)11-s + (−0.423 + 0.906i)13-s + (1.09 − 1.15i)14-s + (−0.104 + 0.180i)16-s + (−0.530 − 0.918i)17-s + (−0.0446 − 0.0119i)19-s + (0.467 − 1.74i)20-s + (1.11 + 1.93i)22-s + (−1.50 − 0.867i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.616 + 0.787i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.616 + 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.77178 - 1.83691i\)
\(L(\frac12)\) \(\approx\) \(3.77178 - 1.83691i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.25 + 1.38i)T \)
13 \( 1 + (1.52 - 3.26i)T \)
good2 \( 1 + (-2.16 + 0.581i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.87 + 1.87i)T - 5iT^{2} \)
11 \( 1 + (-1.20 - 4.51i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.18 + 3.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.194 + 0.0521i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (7.20 + 4.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.20 + 9.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.75 - 6.75i)T - 31iT^{2} \)
37 \( 1 + (-1.22 - 4.58i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.136 + 0.508i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.49 + 1.44i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.928 + 0.928i)T + 47iT^{2} \)
53 \( 1 + 1.95T + 53T^{2} \)
59 \( 1 + (1.76 - 6.57i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.13 + 1.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.40 - 0.377i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.44 + 5.37i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.75 - 8.75i)T + 73iT^{2} \)
79 \( 1 + 4.46T + 79T^{2} \)
83 \( 1 + (-5.42 + 5.42i)T - 83iT^{2} \)
89 \( 1 + (15.0 - 4.03i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.84 + 0.763i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.19979461037422491821159963084, −9.488260997867610779293573171552, −8.475025021797707153962807392219, −7.19687436117249777684233804289, −6.39092251457186885343983493556, −5.27929857902016438313457254913, −4.55637284585218277200204163348, −4.19785671767782954684455421402, −2.35482581857360741890638992515, −1.69204749859234150216263882996, 2.04356610113359644474986800549, 3.07459561143621777973294791143, 4.00097870916107472896080794279, 5.29280417506395427629515230046, 5.85364175081895406435063821569, 6.39783250569954406947080337199, 7.55547313904728809576851821495, 8.449838118775253232404157856701, 9.573480364381842471596771036109, 10.74626214363239341578655874620

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