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Properties

Label 2-720-16.5-c1-0-39
Degree $2$
Conductor $720$
Sign $-0.984 - 0.175i$
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  + (0.720 − 1.21i)2-s + (−0.960 − 1.75i)4-s + (−0.707 − 0.707i)5-s − 0.0588i·7-s + (−2.82 − 0.0955i)8-s + (−1.37 + 0.350i)10-s + (−2.23 − 2.23i)11-s + (2.84 − 2.84i)13-s + (−0.0716 − 0.0424i)14-s + (−2.15 + 3.37i)16-s − 5.98·17-s + (−0.617 + 0.617i)19-s + (−0.561 + 1.91i)20-s + (−4.32 + 1.10i)22-s − 0.746i·23-s + ⋯
L(s)  = 1  + (0.509 − 0.860i)2-s + (−0.480 − 0.877i)4-s + (−0.316 − 0.316i)5-s − 0.0222i·7-s + (−0.999 − 0.0337i)8-s + (−0.433 + 0.110i)10-s + (−0.673 − 0.673i)11-s + (0.790 − 0.790i)13-s + (−0.0191 − 0.0113i)14-s + (−0.538 + 0.842i)16-s − 1.45·17-s + (−0.141 + 0.141i)19-s + (−0.125 + 0.429i)20-s + (−0.922 + 0.236i)22-s − 0.155i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.984 - 0.175i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.984 - 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0956670 + 1.08455i\)
\(L(\frac12)\) \(\approx\) \(0.0956670 + 1.08455i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.720 + 1.21i)T \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + 0.0588iT - 7T^{2} \)
11 \( 1 + (2.23 + 2.23i)T + 11iT^{2} \)
13 \( 1 + (-2.84 + 2.84i)T - 13iT^{2} \)
17 \( 1 + 5.98T + 17T^{2} \)
19 \( 1 + (0.617 - 0.617i)T - 19iT^{2} \)
23 \( 1 + 0.746iT - 23T^{2} \)
29 \( 1 + (-1.13 + 1.13i)T - 29iT^{2} \)
31 \( 1 + 8.55T + 31T^{2} \)
37 \( 1 + (2.01 + 2.01i)T + 37iT^{2} \)
41 \( 1 + 7.71iT - 41T^{2} \)
43 \( 1 + (2.94 + 2.94i)T + 43iT^{2} \)
47 \( 1 - 0.789T + 47T^{2} \)
53 \( 1 + (-6.80 - 6.80i)T + 53iT^{2} \)
59 \( 1 + (-9.36 - 9.36i)T + 59iT^{2} \)
61 \( 1 + (0.814 - 0.814i)T - 61iT^{2} \)
67 \( 1 + (-5.46 + 5.46i)T - 67iT^{2} \)
71 \( 1 + 7.40iT - 71T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 + 17.4T + 79T^{2} \)
83 \( 1 + (-7.55 + 7.55i)T - 83iT^{2} \)
89 \( 1 + 16.3iT - 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−10.42622840470810953648686071455, −8.942600022821482731848152928375, −8.655171424920784246711883200924, −7.34693695230700206954266616594, −6.04162075624248962054242755974, −5.34535364831909957180995986577, −4.22846732903782725596341652863, −3.35283311770297176960551193636, −2.12448734105544034120426337444, −0.45876338124953100182592005182, 2.32334202825989837201461102126, 3.69720049445472002402887640655, 4.52227885721689164763796132863, 5.50753143929936564822579816295, 6.68768403172986290010235074751, 7.07937940723755784512124562205, 8.222821245129718368921599091731, 8.863430107858221302948549417227, 9.866605836498330651674770447348, 11.08756597410004121949779795582

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