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Properties

Label 2-720-9.4-c1-0-11
Degree $2$
Conductor $720$
Sign $0.939 + 0.342i$
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  + (0.866 − 1.5i)3-s + (0.5 − 0.866i)5-s + (1.86 + 3.23i)7-s + (−1.5 − 2.59i)9-s + (2.73 + 4.73i)11-s + (−0.732 + 1.26i)13-s + (−0.866 − 1.5i)15-s + 7.46·17-s + 2·19-s + 6.46·21-s + (0.133 − 0.232i)23-s + (−0.499 − 0.866i)25-s − 5.19·27-s + (−4.23 − 7.33i)29-s + (−1 + 1.73i)31-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + (0.223 − 0.387i)5-s + (0.705 + 1.22i)7-s + (−0.5 − 0.866i)9-s + (0.823 + 1.42i)11-s + (−0.203 + 0.351i)13-s + (−0.223 − 0.387i)15-s + 1.81·17-s + 0.458·19-s + 1.41·21-s + (0.0279 − 0.0483i)23-s + (−0.0999 − 0.173i)25-s − 1.00·27-s + (−0.785 − 1.36i)29-s + (−0.179 + 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.939 + 0.342i$
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: \((2,\ 720,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05605 - 0.362538i\)
\(L(\frac12)\) \(\approx\) \(2.05605 - 0.362538i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (-1.86 - 3.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.73 - 4.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.732 - 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.46T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-0.133 + 0.232i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.23 + 7.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 + (-1.96 + 3.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.73 + 9.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.86 - 3.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (3.19 - 5.53i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.767 - 1.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.86 + 8.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 + 6.92T + 73T^{2} \)
79 \( 1 + (-4.26 - 7.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.40 + 2.42i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.92T + 89T^{2} \)
97 \( 1 + (2.46 + 4.26i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.08887260690458028287075406722, −9.308630420573988560475775394719, −8.695693994116082665355040757781, −7.73072567272829682921404345443, −7.05748822572469880368412497714, −5.85377577678537829012567189287, −5.11656528281010800229079060664, −3.72267651682755436128150591237, −2.26106290758426829005800491590, −1.52129200964677987526090258954, 1.30804776938713247561657938505, 3.26373479817536228400367575992, 3.65573702992317032300555882914, 4.98745926398486597457892330879, 5.80359888194190095953867301198, 7.17490505221813749994428940928, 7.930466823552962759373156909350, 8.752960157459792717080948476050, 9.755304587045957858813528591802, 10.42298664242185935633572009207

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