Properties

Label 2-819-91.45-c1-0-5
Degree $2$
Conductor $819$
Sign $-0.621 - 0.783i$
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  + (0.193 − 0.193i)2-s + 1.92i·4-s + (0.383 − 1.43i)5-s + (−2.15 − 1.53i)7-s + (0.758 + 0.758i)8-s + (−0.202 − 0.350i)10-s + (−1.18 + 4.43i)11-s + (−2.80 + 2.26i)13-s + (−0.713 + 0.119i)14-s − 3.55·16-s − 2.34·17-s + (−1.63 + 0.438i)19-s + (2.75 + 0.737i)20-s + (0.627 + 1.08i)22-s + 4.79i·23-s + ⋯
L(s)  = 1  + (0.136 − 0.136i)2-s + 0.962i·4-s + (0.171 − 0.639i)5-s + (−0.814 − 0.580i)7-s + (0.268 + 0.268i)8-s + (−0.0639 − 0.110i)10-s + (−0.358 + 1.33i)11-s + (−0.778 + 0.627i)13-s + (−0.190 + 0.0319i)14-s − 0.889·16-s − 0.567·17-s + (−0.375 + 0.100i)19-s + (0.615 + 0.164i)20-s + (0.133 + 0.231i)22-s + 0.999i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.621 - 0.783i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349033 + 0.722556i\)
\(L(\frac12)\) \(\approx\) \(0.349033 + 0.722556i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 + 1.53i)T \)
13 \( 1 + (2.80 - 2.26i)T \)
good2 \( 1 + (-0.193 + 0.193i)T - 2iT^{2} \)
5 \( 1 + (-0.383 + 1.43i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.18 - 4.43i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 2.34T + 17T^{2} \)
19 \( 1 + (1.63 - 0.438i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 4.79iT - 23T^{2} \)
29 \( 1 + (-2.87 + 4.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.69 - 1.52i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-6.03 - 6.03i)T + 37iT^{2} \)
41 \( 1 + (-0.829 + 0.222i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.70 + 0.981i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.75 + 2.07i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.54 - 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.09 - 1.09i)T - 59iT^{2} \)
61 \( 1 + (8.45 + 4.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.526 - 0.141i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.77 + 0.474i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.611 + 2.28i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.13 + 3.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.88 + 3.88i)T + 83iT^{2} \)
89 \( 1 + (-9.92 + 9.92i)T - 89iT^{2} \)
97 \( 1 + (-0.734 + 2.73i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.45354208982260281945808847435, −9.540973712528288175911081011990, −9.050101820412598178971890871860, −7.78457882989871558436423254981, −7.26227774957295701968229813792, −6.35430189879704039444571472236, −4.80355711996888532847501043112, −4.34112280605442531767380568468, −3.10935266138098177586530421805, −1.92876157693964602409473180110, 0.35107545858037669338339084292, 2.35455299816667567360803361401, 3.19492913897883338334779633609, 4.72249980412947199681560565866, 5.71144647322531072910547522942, 6.30461428193452580226337823273, 7.05332658224136399007140470835, 8.362537355683941401339911429400, 9.189830930488867503078045936553, 10.01783270470933401925185844484

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