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Properties

Label 2-819-91.16-c1-0-38
Degree $2$
Conductor $819$
Sign $-0.224 + 0.974i$
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  + 0.831·2-s − 1.30·4-s + (1.30 − 2.26i)5-s + (1.78 − 1.95i)7-s − 2.75·8-s + (1.08 − 1.88i)10-s + (−0.924 + 1.60i)11-s + (2.74 − 2.33i)13-s + (1.48 − 1.62i)14-s + 0.331·16-s − 6.83·17-s + (−2.53 − 4.39i)19-s + (−1.71 + 2.96i)20-s + (−0.768 + 1.33i)22-s + 1.27·23-s + ⋯
L(s)  = 1  + 0.587·2-s − 0.654·4-s + (0.585 − 1.01i)5-s + (0.673 − 0.738i)7-s − 0.972·8-s + (0.343 − 0.595i)10-s + (−0.278 + 0.482i)11-s + (0.761 − 0.648i)13-s + (0.396 − 0.434i)14-s + 0.0828·16-s − 1.65·17-s + (−0.581 − 1.00i)19-s + (−0.382 + 0.663i)20-s + (−0.163 + 0.283i)22-s + 0.265·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.224 + 0.974i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.224 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02776 - 1.29088i\)
\(L(\frac12)\) \(\approx\) \(1.02776 - 1.29088i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 + 1.95i)T \)
13 \( 1 + (-2.74 + 2.33i)T \)
good2 \( 1 - 0.831T + 2T^{2} \)
5 \( 1 + (-1.30 + 2.26i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.924 - 1.60i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 6.83T + 17T^{2} \)
19 \( 1 + (2.53 + 4.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.27T + 23T^{2} \)
29 \( 1 + (0.724 + 1.25i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.09 + 5.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.87T + 37T^{2} \)
41 \( 1 + (4.41 + 7.64i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.109 + 0.189i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.624 - 1.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.33 + 2.32i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + (-4.36 - 7.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.91 - 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.78 - 3.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.26 + 5.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.08 + 5.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.67T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + (-6.08 + 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.940679125666282962179736859135, −8.910925648503672114533715428142, −8.625100642801260557023658451559, −7.44242926092027375769527066492, −6.25546183890898644124012751239, −5.27189036085553040138686758386, −4.62251189024610913467571308324, −3.92856854603012248700076718945, −2.25037471218461630076244780905, −0.68631474401703797683411106656, 1.96028355474326836800778077426, 3.06361239233489320521787942277, 4.18098005851627096319991007531, 5.14516687859610520420228929715, 6.12737686471259875034507243044, 6.60385944099583102109531610992, 8.142227285335248049739042081414, 8.777835266427657164831118045719, 9.539506510268919448959758993936, 10.68878982760480954764332312765

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