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Properties

Label 2-819-91.59-c1-0-4
Degree $2$
Conductor $819$
Sign $-0.116 - 0.993i$
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  + (−0.203 − 0.203i)2-s − 1.91i·4-s + (−0.499 − 0.133i)5-s + (−2.60 + 0.449i)7-s + (−0.798 + 0.798i)8-s + (0.0744 + 0.129i)10-s + (−3.69 − 0.990i)11-s + (1.12 + 3.42i)13-s + (0.622 + 0.439i)14-s − 3.50·16-s + 4.52·17-s + (0.797 + 2.97i)19-s + (−0.256 + 0.957i)20-s + (0.551 + 0.955i)22-s + 8.67i·23-s + ⋯
L(s)  = 1  + (−0.144 − 0.144i)2-s − 0.958i·4-s + (−0.223 − 0.0598i)5-s + (−0.985 + 0.169i)7-s + (−0.282 + 0.282i)8-s + (0.0235 + 0.0407i)10-s + (−1.11 − 0.298i)11-s + (0.312 + 0.950i)13-s + (0.166 + 0.117i)14-s − 0.877·16-s + 1.09·17-s + (0.183 + 0.683i)19-s + (−0.0573 + 0.214i)20-s + (0.117 + 0.203i)22-s + 1.80i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.116 - 0.993i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.116 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.290909 + 0.327072i\)
\(L(\frac12)\) \(\approx\) \(0.290909 + 0.327072i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 - 0.449i)T \)
13 \( 1 + (-1.12 - 3.42i)T \)
good2 \( 1 + (0.203 + 0.203i)T + 2iT^{2} \)
5 \( 1 + (0.499 + 0.133i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.69 + 0.990i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 4.52T + 17T^{2} \)
19 \( 1 + (-0.797 - 2.97i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 8.67iT - 23T^{2} \)
29 \( 1 + (1.26 - 2.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.270 - 1.00i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.129 - 0.129i)T - 37iT^{2} \)
41 \( 1 + (1.79 + 6.68i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-4.59 + 2.65i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.42 - 9.03i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.512 - 0.887i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.368 + 0.368i)T + 59iT^{2} \)
61 \( 1 + (7.39 + 4.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.83 - 6.83i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.96 - 7.33i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (11.9 - 3.20i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.77 - 6.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.42 - 6.42i)T - 83iT^{2} \)
89 \( 1 + (8.56 + 8.56i)T + 89iT^{2} \)
97 \( 1 + (13.1 + 3.53i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.28279148341107706323712218378, −9.726793912144743553221797601959, −9.021280641895846152029657268776, −7.909385822207405154391161105316, −6.99972105086916449654745497435, −5.80889720644595718041972432606, −5.51496143206270784322919948698, −4.02270553176448097374321385179, −2.92057932730585942885782741212, −1.51007600477506528704284996431, 0.22382907236741924267825273044, 2.69160130246963188048726357507, 3.31735404277389257020601114318, 4.46308244968687435777335241994, 5.69074274792269672235506034025, 6.68356845444480590996742411649, 7.63753048615859062832968877385, 8.071378026314273840579371547045, 9.114032170529451401398146333918, 10.03571470453299422476013651621

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