Properties

Label 2-2736-76.31-c1-0-14
Degree $2$
Conductor $2736$
Sign $0.952 - 0.305i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + 2i·7-s − 2i·11-s + (−1.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (1.73 + 4i)19-s + (0.866 + 0.5i)23-s + (2 − 3.46i)25-s + (1.5 + 0.866i)29-s − 3.46·31-s + (1.73 − i)35-s + 3.46i·37-s + (−4.5 + 2.59i)41-s + (7.79 − 4.5i)43-s + (9.52 + 5.5i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + 0.755i·7-s − 0.603i·11-s + (−0.416 − 0.240i)13-s + (0.121 + 0.210i)17-s + (0.397 + 0.917i)19-s + (0.180 + 0.104i)23-s + (0.400 − 0.692i)25-s + (0.278 + 0.160i)29-s − 0.622·31-s + (0.292 − 0.169i)35-s + 0.569i·37-s + (−0.702 + 0.405i)41-s + (1.18 − 0.686i)43-s + (1.38 + 0.802i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.952 - 0.305i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.952 - 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.632692141\)
\(L(\frac12)\) \(\approx\) \(1.632692141\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (-1.73 - 4i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 - 3.46iT - 37T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.79 + 4.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.52 - 5.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.5 + 4.33i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.33 - 7.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.59 + 4.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.79 - 13.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 + (-13.5 - 7.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.5 - 7.79i)T + (48.5 - 84.0i)T^{2} \)
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

−8.769225004677905628196884913470, −8.210541967857736156764399547082, −7.48470239466300338638086910517, −6.48426871394514074481996854472, −5.68681027450689196882613956309, −5.10901088863053932715570219792, −4.07648523359913160380116721928, −3.16823521032454024210876061760, −2.20476987571863987148729585213, −0.890298257165407281855597979795, 0.71719748984312550499089244316, 2.10880101523515606431501660447, 3.11184663614998544006087200731, 4.03684102087668396967238404704, 4.79829773390585926101805701546, 5.64056174685023744387040549497, 6.84496340477391857733181661092, 7.17408267018293847329822615871, 7.80552511459070888497457933254, 8.931037779697432124777504963780

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