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Properties

Label 2-720-9.4-c1-0-4
Degree $2$
Conductor $720$
Sign $0.156 - 0.987i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.62 + 0.606i)3-s + (0.5 − 0.866i)5-s + (2.05 + 3.55i)7-s + (2.26 − 1.96i)9-s + (−1.90 − 3.30i)11-s + (−2.90 + 5.03i)13-s + (−0.285 + 1.70i)15-s + 3.81·17-s + 1.81·19-s + (−5.48 − 4.51i)21-s + (1.05 − 1.81i)23-s + (−0.499 − 0.866i)25-s + (−2.48 + 4.56i)27-s + (3.60 + 6.23i)29-s + (−0.908 + 1.57i)31-s + ⋯
L(s)  = 1  + (−0.936 + 0.350i)3-s + (0.223 − 0.387i)5-s + (0.774 + 1.34i)7-s + (0.754 − 0.655i)9-s + (−0.575 − 0.996i)11-s + (−0.806 + 1.39i)13-s + (−0.0738 + 0.441i)15-s + 0.925·17-s + 0.416·19-s + (−1.19 − 0.985i)21-s + (0.219 − 0.379i)23-s + (−0.0999 − 0.173i)25-s + (−0.477 + 0.878i)27-s + (0.668 + 1.15i)29-s + (−0.163 + 0.282i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.156 - 0.987i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.156 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.832693 + 0.711081i\)
\(L(\frac12)\) \(\approx\) \(0.832693 + 0.711081i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 0.606i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (-2.05 - 3.55i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.90 + 3.30i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.90 - 5.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.81T + 17T^{2} \)
19 \( 1 - 1.81T + 19T^{2} \)
23 \( 1 + (-1.05 + 1.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.60 - 6.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.908 - 1.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.01T + 37T^{2} \)
41 \( 1 + (5.50 - 9.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.90 - 5.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.95 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.20T + 53T^{2} \)
59 \( 1 + (2.10 - 3.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.50 - 2.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.85 + 3.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.01T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.94 + 3.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (6.10 + 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.74551489372520532137668241671, −9.696836258033446767700651873784, −8.993717741121440638650415203852, −8.174538023872338923717019631113, −6.94364065081386877338033607607, −5.88394552061530630227553756764, −5.23579974646912923412489508482, −4.57406373666439428331389648256, −2.94348043339359222853253766629, −1.44972781845619277048698071395, 0.69313628937805168665240632945, 2.16512470437810504727440073819, 3.80609322285335870522135891233, 5.04332127379875022774943716349, 5.49287142553635404648856262749, 6.98640112116287855740244230633, 7.42393687776272214295419107171, 8.061726916448220739519319045583, 9.943468276829220024056788412114, 10.25771693116680373132472002649

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