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Properties

Label 2-720-16.5-c1-0-0
Degree $2$
Conductor $720$
Sign $-0.946 + 0.321i$
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  + (−0.562 + 1.29i)2-s + (−1.36 − 1.45i)4-s + (−0.707 − 0.707i)5-s − 1.73i·7-s + (2.66 − 0.952i)8-s + (1.31 − 0.519i)10-s + (−0.505 − 0.505i)11-s + (−1.88 + 1.88i)13-s + (2.25 + 0.977i)14-s + (−0.262 + 3.99i)16-s − 4.53·17-s + (−3.22 + 3.22i)19-s + (−0.0655 + 1.99i)20-s + (0.940 − 0.371i)22-s + 8.85i·23-s + ⋯
L(s)  = 1  + (−0.397 + 0.917i)2-s + (−0.683 − 0.729i)4-s + (−0.316 − 0.316i)5-s − 0.656i·7-s + (0.941 − 0.336i)8-s + (0.415 − 0.164i)10-s + (−0.152 − 0.152i)11-s + (−0.523 + 0.523i)13-s + (0.602 + 0.261i)14-s + (−0.0655 + 0.997i)16-s − 1.09·17-s + (−0.738 + 0.738i)19-s + (−0.0146 + 0.446i)20-s + (0.200 − 0.0791i)22-s + 1.84i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.946 + 0.321i$
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: \((2,\ 720,\ (\ :1/2),\ -0.946 + 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0267533 - 0.162111i\)
\(L(\frac12)\) \(\approx\) \(0.0267533 - 0.162111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.562 - 1.29i)T \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + (0.505 + 0.505i)T + 11iT^{2} \)
13 \( 1 + (1.88 - 1.88i)T - 13iT^{2} \)
17 \( 1 + 4.53T + 17T^{2} \)
19 \( 1 + (3.22 - 3.22i)T - 19iT^{2} \)
23 \( 1 - 8.85iT - 23T^{2} \)
29 \( 1 + (-2.44 + 2.44i)T - 29iT^{2} \)
31 \( 1 + 5.70T + 31T^{2} \)
37 \( 1 + (5.35 + 5.35i)T + 37iT^{2} \)
41 \( 1 - 10.0iT - 41T^{2} \)
43 \( 1 + (2.10 + 2.10i)T + 43iT^{2} \)
47 \( 1 + 4.32T + 47T^{2} \)
53 \( 1 + (-1.37 - 1.37i)T + 53iT^{2} \)
59 \( 1 + (6.64 + 6.64i)T + 59iT^{2} \)
61 \( 1 + (-5.26 + 5.26i)T - 61iT^{2} \)
67 \( 1 + (10.5 - 10.5i)T - 67iT^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 - 6.63iT - 73T^{2} \)
79 \( 1 - 4.27T + 79T^{2} \)
83 \( 1 + (9.15 - 9.15i)T - 83iT^{2} \)
89 \( 1 - 3.23iT - 89T^{2} \)
97 \( 1 - 1.94T + 97T^{2} \)
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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

−10.73122836408138112697711739594, −9.802604547614229294227753653692, −9.085543977137413071694913634567, −8.188384237335917383337872445615, −7.42287339289532154423587878418, −6.68371266717257491895795302155, −5.62853066020826111427530876716, −4.61962587111267249643547680080, −3.78091139644250926956626292317, −1.69761644175826001047838564569, 0.094352677210993824183312966471, 2.15816220150146984253238913577, 2.93089729971491002627717125454, 4.26425274267991626325698741908, 5.09794318684577358944838281475, 6.56488152585703350841273718513, 7.46254617547039869100690173147, 8.704883380079740933178423127220, 8.838624085063713042157832055943, 10.23658388633881310996616226583

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