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Properties

Label 2-819-91.4-c1-0-7
Degree $2$
Conductor $819$
Sign $-0.617 + 0.786i$
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  + 1.73i·2-s − 0.999·4-s + (−1.5 + 0.866i)5-s + (−2 + 1.73i)7-s + 1.73i·8-s + (−1.49 − 2.59i)10-s + (4.5 − 2.59i)11-s + (−1 + 3.46i)13-s + (−2.99 − 3.46i)14-s − 5·16-s − 6·17-s + (−1.5 − 0.866i)19-s + (1.49 − 0.866i)20-s + (4.5 + 7.79i)22-s + (−1 + 1.73i)25-s + (−5.99 − 1.73i)26-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.499·4-s + (−0.670 + 0.387i)5-s + (−0.755 + 0.654i)7-s + 0.612i·8-s + (−0.474 − 0.821i)10-s + (1.35 − 0.783i)11-s + (−0.277 + 0.960i)13-s + (−0.801 − 0.925i)14-s − 1.25·16-s − 1.45·17-s + (−0.344 − 0.198i)19-s + (0.335 − 0.193i)20-s + (0.959 + 1.66i)22-s + (−0.200 + 0.346i)25-s + (−1.17 − 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.617 + 0.786i$
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: \((2,\ 819,\ (\ :1/2),\ -0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319136 - 0.655852i\)
\(L(\frac12)\) \(\approx\) \(0.319136 - 0.655852i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.5 - 4.33i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.46iT - 59T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.5 - 0.866i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.06709049845491620878135456073, −9.461729995637574624587509159443, −8.948453446334309863323813209684, −8.190635449512192185302554112814, −7.01357712979674356015664310878, −6.58776826406679042434371713455, −5.93909252172588897017274096581, −4.62148873339262435083459410236, −3.63732330252182250802151408991, −2.26337023830520969106912709463, 0.34478588260048123465662173795, 1.75589269891901704197014499739, 3.12368173493496481052593583461, 4.00596738579580187418725283472, 4.64132726134041391664063347230, 6.44839452299135981570620326070, 6.96341754018864267456351016017, 8.135604531706355311467145654195, 9.208899204394882823047853630766, 9.816880917378445532088554209020

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