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Properties

Label 2-819-91.74-c1-0-27
Degree $2$
Conductor $819$
Sign $-0.119 + 0.992i$
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  − 1.85·2-s + 1.45·4-s + (−0.0986 − 0.170i)5-s + (1.03 + 2.43i)7-s + 1.01·8-s + (0.183 + 0.317i)10-s + (−2.09 − 3.62i)11-s + (−2.72 − 2.36i)13-s + (−1.92 − 4.52i)14-s − 4.79·16-s − 0.841·17-s + (−0.675 + 1.17i)19-s + (−0.143 − 0.248i)20-s + (3.88 + 6.73i)22-s + 4.11·23-s + ⋯
L(s)  = 1  − 1.31·2-s + 0.726·4-s + (−0.0441 − 0.0764i)5-s + (0.392 + 0.919i)7-s + 0.359·8-s + (0.0579 + 0.100i)10-s + (−0.630 − 1.09i)11-s + (−0.755 − 0.655i)13-s + (−0.515 − 1.20i)14-s − 1.19·16-s − 0.204·17-s + (−0.155 + 0.268i)19-s + (−0.0320 − 0.0555i)20-s + (0.828 + 1.43i)22-s + 0.858·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.119 + 0.992i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.119 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.302876 - 0.341540i\)
\(L(\frac12)\) \(\approx\) \(0.302876 - 0.341540i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.03 - 2.43i)T \)
13 \( 1 + (2.72 + 2.36i)T \)
good2 \( 1 + 1.85T + 2T^{2} \)
5 \( 1 + (0.0986 + 0.170i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.09 + 3.62i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.841T + 17T^{2} \)
19 \( 1 + (0.675 - 1.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.11T + 23T^{2} \)
29 \( 1 + (4.11 - 7.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.640 + 1.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.04T + 37T^{2} \)
41 \( 1 + (-2.69 + 4.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.66 + 4.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.83 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.32 + 4.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.05T + 59T^{2} \)
61 \( 1 + (-5.68 + 9.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.69 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.98 + 5.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.94 - 3.36i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.36 - 9.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.07T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + (9.73 + 16.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.911613795097021786916925685668, −8.971687716402142702426721241767, −8.468333366386059624022044467818, −7.80831792975401349013222884779, −6.82664648836786625660129370423, −5.58152493529246270259286032505, −4.84185487784998570084999696489, −3.13765727424324796419033677463, −1.99350922290984693771327939166, −0.37595699925707566388206069861, 1.29989463111603649165655290291, 2.52063404593312267626766627345, 4.31763776446987871022036864720, 4.89505840949075421081473030872, 6.58286073775338164675612153609, 7.52499741741304368367160015390, 7.66131430551701953131909316940, 8.938645666206683983555753537603, 9.622075798506264526712070150004, 10.24328662717679263122872868634

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