↑

Properties

Label 2-2736-19.18-c2-0-49
Degree $2$
Conductor $2736$
Sign $0.629 - 0.776i$
Analytic cond. $74.5506$
Root an. cond. $8.63426$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.08·5-s + 11.6·7-s + 14.6·11-s + 16.9i·13-s + 8.79·17-s + (11.9 − 14.7i)19-s + 6.84·23-s + 0.900·25-s + 45.8i·29-s + 50.1i·31-s − 59.5·35-s + 35.1i·37-s − 60.4i·41-s + 75.0·43-s − 52.6·47-s + ⋯
L(s)  = 1  − 1.01·5-s + 1.67·7-s + 1.33·11-s + 1.30i·13-s + 0.517·17-s + (0.629 − 0.776i)19-s + 0.297·23-s + 0.0360·25-s + 1.57i·29-s + 1.61i·31-s − 1.70·35-s + 0.951i·37-s − 1.47i·41-s + 1.74·43-s − 1.12·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.629 - 0.776i$
Analytic conductor: \(74.5506\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ 0.629 - 0.776i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.422863820\)
\(L(\frac12)\) \(\approx\) \(2.422863820\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (-11.9 + 14.7i)T \)
good5 \( 1 + 5.08T + 25T^{2} \)
7 \( 1 - 11.6T + 49T^{2} \)
11 \( 1 - 14.6T + 121T^{2} \)
13 \( 1 - 16.9iT - 169T^{2} \)
17 \( 1 - 8.79T + 289T^{2} \)
23 \( 1 - 6.84T + 529T^{2} \)
29 \( 1 - 45.8iT - 841T^{2} \)
31 \( 1 - 50.1iT - 961T^{2} \)
37 \( 1 - 35.1iT - 1.36e3T^{2} \)
41 \( 1 + 60.4iT - 1.68e3T^{2} \)
43 \( 1 - 75.0T + 1.84e3T^{2} \)
47 \( 1 + 52.6T + 2.20e3T^{2} \)
53 \( 1 + 93.2iT - 2.80e3T^{2} \)
59 \( 1 + 13.0iT - 3.48e3T^{2} \)
61 \( 1 + 1.36T + 3.72e3T^{2} \)
67 \( 1 - 69.4iT - 4.48e3T^{2} \)
71 \( 1 - 58.7iT - 5.04e3T^{2} \)
73 \( 1 + 41.4T + 5.32e3T^{2} \)
79 \( 1 + 90.3iT - 6.24e3T^{2} \)
83 \( 1 + 155.T + 6.88e3T^{2} \)
89 \( 1 + 73.4iT - 7.92e3T^{2} \)
97 \( 1 - 36.7iT - 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.777934280282870545206714271775, −8.056121008163617305603128070881, −7.08700669880907349135939927178, −6.89411742344680235667642717859, −5.44947397143010705150510995634, −4.71612121014775516273035818989, −4.11290071739429172489966600184, −3.25454104181230536561583897547, −1.76540485701384959676002270319, −1.10385470270614744204136383883, 0.68071335451519600193094066524, 1.54284707481672771465989172032, 2.85417692641953132566721569966, 4.03546868969439800604884490434, 4.31808394580382569785282588679, 5.49061940576618016277130844932, 6.07107598085866029254086495945, 7.43534278797285415613425960929, 7.87402185456482753758571112059, 8.168471788750489810930976453791

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