↑

Properties

Label 2-720-5.4-c5-0-57
Degree $2$
Conductor $720$
Sign $-0.235 + 0.971i$
Analytic cond. $115.476$
Root an. cond. $10.7459$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (13.1 − 54.3i)5-s + 146. i·7-s + 191.·11-s + 83.9i·13-s − 2.00e3i·17-s + 677.·19-s + 1.29e3i·23-s + (−2.77e3 − 1.42e3i)25-s − 3.26e3·29-s − 6.15e3·31-s + (7.97e3 + 1.93e3i)35-s + 1.13e4i·37-s + 1.05e4·41-s − 1.29e4i·43-s − 9.52e3i·47-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)5-s + 1.13i·7-s + 0.476·11-s + 0.137i·13-s − 1.67i·17-s + 0.430·19-s + 0.510i·23-s + (−0.889 − 0.457i)25-s − 0.721·29-s − 1.15·31-s + (1.10 + 0.266i)35-s + 1.36i·37-s + 0.984·41-s − 1.06i·43-s − 0.628i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.235 + 0.971i$
Analytic conductor: \(115.476\)
Root analytic conductor: \(10.7459\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :5/2),\ -0.235 + 0.971i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.646412068\)
\(L(\frac12)\) \(\approx\) \(1.646412068\)
\(L(\frac{7}{2})\) 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 \( 1 \)
5 \( 1 + (-13.1 + 54.3i)T \)
good7 \( 1 - 146. iT - 1.68e4T^{2} \)
11 \( 1 - 191.T + 1.61e5T^{2} \)
13 \( 1 - 83.9iT - 3.71e5T^{2} \)
17 \( 1 + 2.00e3iT - 1.41e6T^{2} \)
19 \( 1 - 677.T + 2.47e6T^{2} \)
23 \( 1 - 1.29e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.26e3T + 2.05e7T^{2} \)
31 \( 1 + 6.15e3T + 2.86e7T^{2} \)
37 \( 1 - 1.13e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.05e4T + 1.15e8T^{2} \)
43 \( 1 + 1.29e4iT - 1.47e8T^{2} \)
47 \( 1 + 9.52e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.47e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.82e4T + 7.14e8T^{2} \)
61 \( 1 + 3.58e3T + 8.44e8T^{2} \)
67 \( 1 - 2.17e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.13e4T + 1.80e9T^{2} \)
73 \( 1 + 1.33e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.59e4T + 3.07e9T^{2} \)
83 \( 1 + 5.33e4iT - 3.93e9T^{2} \)
89 \( 1 + 5.13e4T + 5.58e9T^{2} \)
97 \( 1 + 8.08e4iT - 8.58e9T^{2} \)
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

−9.234798881905663620289802308134, −8.829338826891346711026325831924, −7.74967700142122574007179509615, −6.74694169187807298907596136619, −5.49956990078726753464623906378, −5.18530626364323829444221298153, −3.89873082930688006727976039435, −2.61658104911642400989684470063, −1.58135566858768221600228768031, −0.36171891910173093535228290110, 1.08711662988951257461182819972, 2.23418420605270386679722770888, 3.59318526703577073335998981585, 4.10335060159638346953169000086, 5.63443166967463630863525127279, 6.44436110684618593924350566270, 7.27925718178122363516563610940, 7.952279362495052154422455909593, 9.189177067266470430787973864274, 10.01107623221740619884121549118

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