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Properties

Label 2-819-91.4-c1-0-40
Degree $2$
Conductor $819$
Sign $-0.381 + 0.924i$
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  − 1.16i·2-s + 0.651·4-s + (3.43 − 1.98i)5-s + (−2.10 + 1.59i)7-s − 3.07i·8-s + (−2.30 − 3.98i)10-s + (0.144 − 0.0836i)11-s + (−0.211 − 3.59i)13-s + (1.85 + 2.44i)14-s − 2.27·16-s − 1.81·17-s + (−3.45 − 1.99i)19-s + (2.23 − 1.29i)20-s + (−0.0971 − 0.168i)22-s + 3.60·23-s + ⋯
L(s)  = 1  − 0.821i·2-s + 0.325·4-s + (1.53 − 0.886i)5-s + (−0.797 + 0.603i)7-s − 1.08i·8-s + (−0.727 − 1.26i)10-s + (0.0437 − 0.0252i)11-s + (−0.0586 − 0.998i)13-s + (0.495 + 0.654i)14-s − 0.567·16-s − 0.439·17-s + (−0.792 − 0.457i)19-s + (0.500 − 0.288i)20-s + (−0.0207 − 0.0358i)22-s + 0.752·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.381 + 0.924i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.381 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16053 - 1.73373i\)
\(L(\frac12)\) \(\approx\) \(1.16053 - 1.73373i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 - 1.59i)T \)
13 \( 1 + (0.211 + 3.59i)T \)
good2 \( 1 + 1.16iT - 2T^{2} \)
5 \( 1 + (-3.43 + 1.98i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.144 + 0.0836i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.81T + 17T^{2} \)
19 \( 1 + (3.45 + 1.99i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.60T + 23T^{2} \)
29 \( 1 + (2.60 - 4.50i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.95 - 1.12i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.31iT - 37T^{2} \)
41 \( 1 + (-9.53 - 5.50i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.64 - 8.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.62 + 4.98i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.26 + 5.65i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.11iT - 59T^{2} \)
61 \( 1 + (6.50 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 - 5.95i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.62 - 2.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.65 + 5.57i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.727 + 1.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.67iT - 83T^{2} \)
89 \( 1 + 7.80iT - 89T^{2} \)
97 \( 1 + (-9.81 + 5.66i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.13365125640976034920937866045, −9.145335548548704807927012695761, −8.855586538083199707360102623520, −7.26661589828547452453938573016, −6.19269074722417266841424033131, −5.71743566435969262589171882523, −4.52606882976944375606097759381, −2.97542811998867031804542108633, −2.30925135374801976192688310937, −1.04497022013865876752228572707, 1.94269088683779073807916734491, 2.78315731965213898946165550690, 4.27077543182456130972271609993, 5.74711964867362471608882066318, 6.22065890051060661507139135354, 6.89679322118389822800685909136, 7.51506495878542064417491762911, 8.980637501649354795987328806065, 9.533701747358206223812947588258, 10.64803655588282333599446096187

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