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Properties

Label 2-819-91.20-c1-0-27
Degree $2$
Conductor $819$
Sign $0.0860 + 0.996i$
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  + (0.112 − 0.0302i)2-s + (−1.72 + 0.993i)4-s + (−1.24 + 1.24i)5-s + (−2.63 + 0.206i)7-s + (−0.329 + 0.329i)8-s + (−0.103 + 0.178i)10-s + (0.506 + 1.89i)11-s + (1.85 − 3.09i)13-s + (−0.291 + 0.102i)14-s + (1.95 − 3.39i)16-s + (−2.13 − 3.70i)17-s + (−4.12 − 1.10i)19-s + (0.907 − 3.38i)20-s + (0.114 + 0.197i)22-s + (5.53 + 3.19i)23-s + ⋯
L(s)  = 1  + (0.0797 − 0.0213i)2-s + (−0.860 + 0.496i)4-s + (−0.558 + 0.558i)5-s + (−0.996 + 0.0779i)7-s + (−0.116 + 0.116i)8-s + (−0.0325 + 0.0564i)10-s + (0.152 + 0.570i)11-s + (0.515 − 0.857i)13-s + (−0.0778 + 0.0275i)14-s + (0.489 − 0.848i)16-s + (−0.518 − 0.898i)17-s + (−0.945 − 0.253i)19-s + (0.202 − 0.757i)20-s + (0.0243 + 0.0422i)22-s + (1.15 + 0.666i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0860 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0860 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.0860 + 0.996i$
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: \((2,\ 819,\ (\ :1/2),\ 0.0860 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363098 - 0.333087i\)
\(L(\frac12)\) \(\approx\) \(0.363098 - 0.333087i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 0.206i)T \)
13 \( 1 + (-1.85 + 3.09i)T \)
good2 \( 1 + (-0.112 + 0.0302i)T + (1.73 - i)T^{2} \)
5 \( 1 + (1.24 - 1.24i)T - 5iT^{2} \)
11 \( 1 + (-0.506 - 1.89i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.13 + 3.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.12 + 1.10i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.53 - 3.19i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.57 + 6.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.02 + 3.02i)T - 31iT^{2} \)
37 \( 1 + (0.732 + 2.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.94 + 11.0i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.55 - 0.896i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.68 + 4.68i)T + 47iT^{2} \)
53 \( 1 + 4.27T + 53T^{2} \)
59 \( 1 + (0.436 - 1.62i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.66 - 1.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0190 - 0.00510i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.23 - 4.59i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.698 + 0.698i)T + 73iT^{2} \)
79 \( 1 + 5.93T + 79T^{2} \)
83 \( 1 + (-9.87 + 9.87i)T - 83iT^{2} \)
89 \( 1 + (7.76 - 2.07i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (14.2 + 3.82i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.921837689156164100721385410079, −9.194940309074629289812425261132, −8.407081375376052815280327655424, −7.40910959705472756259659292172, −6.73850219609901063293997908927, −5.54858751941279748753243942485, −4.43936040544215543280188973435, −3.53695626238872910036158029560, −2.73063990354781548743380774798, −0.27545360348362169697022154274, 1.23049652467201820214522008640, 3.21289768144149940066218786269, 4.21414835726830512467694130050, 4.86489955263358007856037276472, 6.29444315393368271537675239952, 6.58546518920539597042256205082, 8.374469289643551827938206490378, 8.618779751599513396749609948301, 9.472153843418807245124766494253, 10.41601758103147408924583072685

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