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Properties

Label 2-720-60.59-c5-0-33
Degree $2$
Conductor $720$
Sign $0.469 + 0.882i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (55.4 + 7.07i)5-s − 168.·7-s − 318.·11-s − 172. i·13-s − 487.·17-s + 1.67e3i·19-s + 2.61e3i·23-s + (3.02e3 + 784. i)25-s − 4.32e3i·29-s + 6.36e3i·31-s + (−9.32e3 − 1.18e3i)35-s − 6.22e3i·37-s − 1.30e4i·41-s + 9.08e3·43-s − 1.04e4i·47-s + ⋯
L(s)  = 1  + (0.991 + 0.126i)5-s − 1.29·7-s − 0.793·11-s − 0.283i·13-s − 0.409·17-s + 1.06i·19-s + 1.03i·23-s + (0.968 + 0.250i)25-s − 0.954i·29-s + 1.19i·31-s + (−1.28 − 0.164i)35-s − 0.747i·37-s − 1.21i·41-s + 0.749·43-s − 0.691i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.469 + 0.882i$
Analytic conductor: \(115.476\)
Root analytic conductor: \(10.7459\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :5/2),\ 0.469 + 0.882i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.412918838\)
\(L(\frac12)\) \(\approx\) \(1.412918838\)
\(L(\frac{7}{2})\) 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 \( 1 + (-55.4 - 7.07i)T \)
good7 \( 1 + 168.T + 1.68e4T^{2} \)
11 \( 1 + 318.T + 1.61e5T^{2} \)
13 \( 1 + 172. iT - 3.71e5T^{2} \)
17 \( 1 + 487.T + 1.41e6T^{2} \)
19 \( 1 - 1.67e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.61e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.32e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.36e3iT - 2.86e7T^{2} \)
37 \( 1 + 6.22e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.30e4iT - 1.15e8T^{2} \)
43 \( 1 - 9.08e3T + 1.47e8T^{2} \)
47 \( 1 + 1.04e4iT - 2.29e8T^{2} \)
53 \( 1 + 7.45e3T + 4.18e8T^{2} \)
59 \( 1 - 8.50e3T + 7.14e8T^{2} \)
61 \( 1 - 4.51e3T + 8.44e8T^{2} \)
67 \( 1 - 1.48e4T + 1.35e9T^{2} \)
71 \( 1 - 4.20e4T + 1.80e9T^{2} \)
73 \( 1 + 3.34e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.08e5iT - 3.07e9T^{2} \)
83 \( 1 - 2.90e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.77e4iT - 5.58e9T^{2} \)
97 \( 1 + 7.97e4iT - 8.58e9T^{2} \)
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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

−9.637116638274785648591275849098, −8.837526573808294679686652731853, −7.68806385692262508890255300012, −6.74677386534912230744235373952, −5.91409103796007292212100444471, −5.27644180567335225152368920413, −3.77147892046099572669780415401, −2.84939493133420177980240311677, −1.84145327068086069108485033738, −0.35626894090749291561244390470, 0.816394752810718545299724086301, 2.33924538210119684732624455950, 2.98332917508725076510199639558, 4.42348832025387526502912591725, 5.39830632352106631380258042100, 6.39351732294314512887620574114, 6.86561916126571966689375832339, 8.187003757327416066763150151980, 9.192748431294849917213820131549, 9.683248498054864280961963368252

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