↑

Properties

Label 2-2736-76.27-c1-0-25
Degree $2$
Conductor $2736$
Sign $0.815 - 0.579i$
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  + (−1.85 + 3.21i)5-s − 2.36i·7-s + 4.10i·11-s + (3 − 1.73i)13-s + (3.91 − 6.77i)17-s + (4.35 + 0.0157i)19-s + (−2.05 + 1.18i)23-s + (−4.41 − 7.63i)25-s + (−2.57 + 1.48i)29-s − 1.33·31-s + (7.62 + 4.40i)35-s − 2.97i·37-s + (−2.07 − 1.20i)41-s + (3 + 1.73i)43-s + (5.05 − 2.91i)47-s + ⋯
L(s)  = 1  + (−0.831 + 1.43i)5-s − 0.895i·7-s + 1.23i·11-s + (0.832 − 0.480i)13-s + (0.948 − 1.64i)17-s + (0.999 + 0.00360i)19-s + (−0.427 + 0.247i)23-s + (−0.882 − 1.52i)25-s + (−0.478 + 0.276i)29-s − 0.239·31-s + (1.28 + 0.744i)35-s − 0.489i·37-s + (−0.324 − 0.187i)41-s + (0.457 + 0.264i)43-s + (0.736 − 0.425i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.815 - 0.579i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.815 - 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.599328213\)
\(L(\frac12)\) \(\approx\) \(1.599328213\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.35 - 0.0157i)T \)
good5 \( 1 + (1.85 - 3.21i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.36iT - 7T^{2} \)
11 \( 1 - 4.10iT - 11T^{2} \)
13 \( 1 + (-3 + 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.91 + 6.77i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.05 - 1.18i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.57 - 1.48i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.33T + 31T^{2} \)
37 \( 1 + 2.97iT - 37T^{2} \)
41 \( 1 + (2.07 + 1.20i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3 - 1.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.05 + 2.91i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 + 3.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.44 - 4.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.141 + 0.244i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.26 - 5.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.306 - 0.531i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.71 - 9.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.0314iT - 83T^{2} \)
89 \( 1 + (-14.7 + 8.50i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.6 - 7.30i)T + (48.5 + 84.0i)T^{2} \)
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

−8.918688126874614632575122268645, −7.62149792319109536954602201517, −7.43394565032901228091280537034, −7.00702689447308197889023735465, −5.89670349270488379746577478506, −4.93778037474099128744997641067, −3.85258129861050234100053505466, −3.39717287036481293400733346065, −2.41361256788897809638381317734, −0.852409231616049791330718855912, 0.77407433373007634741124721936, 1.73988246051560653758980023748, 3.32598753711358750049616811647, 3.88407697549564506322161935348, 4.86044399914334972811246063764, 5.83880180194961753080168071873, 6.01190122908576010519055274808, 7.57054790379603058317471932783, 8.155406534086638521666343712250, 8.777554729666849883364844856873

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