↑

Properties

Label 2-2736-228.83-c1-0-29
Degree $2$
Conductor $2736$
Sign $0.0898 + 0.995i$
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  + (1.45 − 0.842i)5-s + 0.172i·7-s − 2.01·11-s + (−0.109 + 0.189i)13-s + (−5.01 + 2.89i)17-s + (3.23 − 2.92i)19-s + (4.27 − 7.40i)23-s + (−1.08 + 1.87i)25-s + (−0.957 − 0.552i)29-s − 8.93i·31-s + (0.145 + 0.251i)35-s − 6.80·37-s + (5.97 − 3.44i)41-s + (4.91 − 2.83i)43-s + (−1.15 + 1.99i)47-s + ⋯
L(s)  = 1  + (0.652 − 0.376i)5-s + 0.0651i·7-s − 0.608·11-s + (−0.0302 + 0.0524i)13-s + (−1.21 + 0.702i)17-s + (0.742 − 0.670i)19-s + (0.891 − 1.54i)23-s + (−0.216 + 0.374i)25-s + (−0.177 − 0.102i)29-s − 1.60i·31-s + (0.0245 + 0.0424i)35-s − 1.11·37-s + (0.932 − 0.538i)41-s + (0.748 − 0.432i)43-s + (−0.168 + 0.291i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.0898 + 0.995i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.0898 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.583648869\)
\(L(\frac12)\) \(\approx\) \(1.583648869\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 + 2.92i)T \)
good5 \( 1 + (-1.45 + 0.842i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.172iT - 7T^{2} \)
11 \( 1 + 2.01T + 11T^{2} \)
13 \( 1 + (0.109 - 0.189i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.01 - 2.89i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.27 + 7.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.957 + 0.552i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.93iT - 31T^{2} \)
37 \( 1 + 6.80T + 37T^{2} \)
41 \( 1 + (-5.97 + 3.44i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.91 + 2.83i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.15 - 1.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.55 - 2.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.14 + 8.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.39 + 9.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.6 - 6.13i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.30 + 12.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.47 + 4.28i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.20 + 3.00i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + (0.701 + 0.404i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.990 + 1.71i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.865352058873288630510728609013, −7.921254765241619830718430631545, −7.10190986859875011957017722407, −6.30638478646994436518650805624, −5.52771591760155037934162191136, −4.80519723880423329596225129492, −3.95309721482352288539119933759, −2.67713022421396890006674304249, −1.97897498596409717375685830797, −0.51737136382410508432694319794, 1.28070406457447042120887368735, 2.43735803472163263373347420548, 3.18665111537701232622303572916, 4.29279617123871390118648991356, 5.33244903677478792711691413733, 5.74785858427175090326307676751, 6.95880500748961175647393569954, 7.22967241752450892976820531114, 8.307090631883868350543330744918, 9.073400922446199672650747031129

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