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Properties

Label 2-720-12.11-c3-0-19
Degree $2$
Conductor $720$
Sign $0.0917 + 0.995i$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5i·5-s − 32.1i·7-s + 20.9·11-s + 58.0·13-s + 40.1i·17-s − 53.6i·19-s − 48.7·23-s − 25·25-s + 136. i·29-s + 52.9i·31-s + 160.·35-s − 194.·37-s − 438. i·41-s − 360. i·43-s − 174.·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 1.73i·7-s + 0.574·11-s + 1.23·13-s + 0.572i·17-s − 0.648i·19-s − 0.441·23-s − 0.200·25-s + 0.873i·29-s + 0.307i·31-s + 0.776·35-s − 0.862·37-s − 1.67i·41-s − 1.27i·43-s − 0.540·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.0917 + 0.995i$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ 0.0917 + 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.870751170\)
\(L(\frac12)\) \(\approx\) \(1.870751170\)
\(L(\frac{5}{2})\) 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 \( 1 - 5iT \)
good7 \( 1 + 32.1iT - 343T^{2} \)
11 \( 1 - 20.9T + 1.33e3T^{2} \)
13 \( 1 - 58.0T + 2.19e3T^{2} \)
17 \( 1 - 40.1iT - 4.91e3T^{2} \)
19 \( 1 + 53.6iT - 6.85e3T^{2} \)
23 \( 1 + 48.7T + 1.21e4T^{2} \)
29 \( 1 - 136. iT - 2.43e4T^{2} \)
31 \( 1 - 52.9iT - 2.97e4T^{2} \)
37 \( 1 + 194.T + 5.06e4T^{2} \)
41 \( 1 + 438. iT - 6.89e4T^{2} \)
43 \( 1 + 360. iT - 7.95e4T^{2} \)
47 \( 1 + 174.T + 1.03e5T^{2} \)
53 \( 1 + 592. iT - 1.48e5T^{2} \)
59 \( 1 - 738.T + 2.05e5T^{2} \)
61 \( 1 + 36.9T + 2.26e5T^{2} \)
67 \( 1 + 637. iT - 3.00e5T^{2} \)
71 \( 1 - 380.T + 3.57e5T^{2} \)
73 \( 1 - 659.T + 3.89e5T^{2} \)
79 \( 1 + 22.1iT - 4.93e5T^{2} \)
83 \( 1 - 451.T + 5.71e5T^{2} \)
89 \( 1 - 1.67e3iT - 7.04e5T^{2} \)
97 \( 1 - 3.17T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.03814721330971044359787319178, −8.907894364904064473020512846774, −8.057790565555854751774218003038, −6.94034861378765345990139394802, −6.63734202246433371966330097809, −5.27498899236248578294971289725, −3.90219945446604455411169504555, −3.58391218819186484451861845396, −1.73846586174050513140417191832, −0.56033741064342034297202277533, 1.27997492547986556410901527848, 2.46065895479063155365522754179, 3.67143037841355155692905236924, 4.87149212593554721660973351927, 5.88786636596329796060593736846, 6.35608976666851796518636202600, 7.893651457279245609100938588024, 8.585906608559507284588499926810, 9.233132860555448773305961299285, 10.00761716010420128110622638299

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