Properties

Label 2-819-273.74-c0-0-1
Degree $2$
Conductor $819$
Sign $-0.410 + 0.911i$
Analytic cond. $0.408734$
Root an. cond. $0.639323$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 1.00·4-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)13-s + (1.22 − 0.707i)14-s − 0.999·16-s − 1.41i·17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s + (−1.22 − 0.707i)26-s + (−0.500 − 0.866i)28-s + (1.22 + 0.707i)29-s + 1.41i·32-s − 2.00·34-s − 37-s + (−1.22 − 0.707i)38-s + ⋯
L(s)  = 1  − 1.41i·2-s − 1.00·4-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)13-s + (1.22 − 0.707i)14-s − 0.999·16-s − 1.41i·17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s + (−1.22 − 0.707i)26-s + (−0.500 − 0.866i)28-s + (1.22 + 0.707i)29-s + 1.41i·32-s − 2.00·34-s − 37-s + (−1.22 − 0.707i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.410 + 0.911i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (620, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ -0.410 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.029783623\)
\(L(\frac12)\) \(\approx\) \(1.029783623\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + 1.41iT - T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.34704394975137153386494664312, −9.419745271965381845394299387195, −8.880467252971510345294615877519, −7.80159259005987348873172886695, −6.71744815663180844223210719724, −5.41428046205412410644214536467, −4.67845296221088674139930790286, −3.24899284705905564046074302253, −2.63935265313141468885693897318, −1.26642902198179722273586619182, 1.80193875955297551872612182684, 3.79948890656025790740259195610, 4.56723456385672906717109969695, 5.69729838169400291796311246324, 6.47433388570937445379467678298, 7.17377937799014902371952014654, 8.258791976304104379901564714450, 8.400248797839034787321723188066, 9.781814032085503847713707512844, 10.57221404271902931027929551787

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