Properties

Label 2-720-9.7-c1-0-0
Degree $2$
Conductor $720$
Sign $-0.195 - 0.980i$
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  + (0.574 − 1.63i)3-s + (−0.5 − 0.866i)5-s + (−2.28 + 3.96i)7-s + (−2.33 − 1.87i)9-s + (−2.29 + 3.97i)11-s + (1.83 + 3.18i)13-s + (−1.70 + 0.319i)15-s − 2.55·17-s − 4.76·19-s + (5.15 + 6.01i)21-s + (−1.28 − 2.22i)23-s + (−0.499 + 0.866i)25-s + (−4.41 + 2.74i)27-s + (−0.956 + 1.65i)29-s + (1.73 + 3.00i)31-s + ⋯
L(s)  = 1  + (0.331 − 0.943i)3-s + (−0.223 − 0.387i)5-s + (−0.864 + 1.49i)7-s + (−0.779 − 0.625i)9-s + (−0.692 + 1.19i)11-s + (0.510 + 0.883i)13-s + (−0.439 + 0.0824i)15-s − 0.619·17-s − 1.09·19-s + (1.12 + 1.31i)21-s + (−0.268 − 0.464i)23-s + (−0.0999 + 0.173i)25-s + (−0.849 + 0.528i)27-s + (−0.177 + 0.307i)29-s + (0.311 + 0.539i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.195 - 0.980i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.404900 + 0.493365i\)
\(L(\frac12)\) \(\approx\) \(0.404900 + 0.493365i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.574 + 1.63i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (2.28 - 3.96i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.29 - 3.97i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.83 - 3.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.55T + 17T^{2} \)
19 \( 1 + 4.76T + 19T^{2} \)
23 \( 1 + (1.28 + 2.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.956 - 1.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.73 - 3.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.46T + 37T^{2} \)
41 \( 1 + (3.32 + 5.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.27 - 5.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.01 + 8.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.03T + 53T^{2} \)
59 \( 1 + (-6.13 - 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.79 - 8.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.66 + 9.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.67T + 71T^{2} \)
73 \( 1 - 3.12T + 73T^{2} \)
79 \( 1 + (6.64 - 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.39 + 9.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.04T + 89T^{2} \)
97 \( 1 + (8.68 - 15.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.66361118418787225857295690099, −9.478107488534968648197065651637, −8.835613112373411897139323445304, −8.255110504891502397896502046994, −7.02063915814359884152725403525, −6.40400187838591859079906821088, −5.44019344718675456855556216314, −4.16948387454864967929762440367, −2.67752746497369973745296403098, −1.95501076575529036979157308906, 0.29487432835679363550315035846, 2.81166550529556518504483550168, 3.61501325399575269225583603431, 4.34623542779671272488951931987, 5.72142617360462019782125465864, 6.56482741935942377628999714783, 7.79762437875426766207832524159, 8.354723436084422307080501444403, 9.506912144813544633800040426349, 10.33277971745242440517883439191

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