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Properties

Label 2-720-20.19-c2-0-19
Degree $2$
Conductor $720$
Sign $0.5 + 0.866i$
Analytic cond. $19.6185$
Root an. cond. $4.42928$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 6.92·7-s − 13.8i·11-s + 24i·13-s − 24i·17-s − 27.7i·19-s + 34.6·23-s + 25·25-s − 10·29-s − 13.8i·31-s − 34.6·35-s − 24i·37-s + 34·41-s − 20.7·43-s + 6.92·47-s + ⋯
L(s)  = 1  + 5-s − 0.989·7-s − 1.25i·11-s + 1.84i·13-s − 1.41i·17-s − 1.45i·19-s + 1.50·23-s + 25-s − 0.344·29-s − 0.446i·31-s − 0.989·35-s − 0.648i·37-s + 0.829·41-s − 0.483·43-s + 0.147·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(19.6185\)
Root analytic conductor: \(4.42928\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1),\ 0.5 + 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.809265228\)
\(L(\frac12)\) \(\approx\) \(1.809265228\)
\(L(2)\) 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 \( 1 \)
5 \( 1 - 5T \)
good7 \( 1 + 6.92T + 49T^{2} \)
11 \( 1 + 13.8iT - 121T^{2} \)
13 \( 1 - 24iT - 169T^{2} \)
17 \( 1 + 24iT - 289T^{2} \)
19 \( 1 + 27.7iT - 361T^{2} \)
23 \( 1 - 34.6T + 529T^{2} \)
29 \( 1 + 10T + 841T^{2} \)
31 \( 1 + 13.8iT - 961T^{2} \)
37 \( 1 + 24iT - 1.36e3T^{2} \)
41 \( 1 - 34T + 1.68e3T^{2} \)
43 \( 1 + 20.7T + 1.84e3T^{2} \)
47 \( 1 - 6.92T + 2.20e3T^{2} \)
53 \( 1 + 48iT - 2.80e3T^{2} \)
59 \( 1 + 13.8iT - 3.48e3T^{2} \)
61 \( 1 - 70T + 3.72e3T^{2} \)
67 \( 1 - 90.0T + 4.48e3T^{2} \)
71 \( 1 + 55.4iT - 5.04e3T^{2} \)
73 \( 1 - 48iT - 5.32e3T^{2} \)
79 \( 1 + 41.5iT - 6.24e3T^{2} \)
83 \( 1 + 90.0T + 6.88e3T^{2} \)
89 \( 1 - 14T + 7.92e3T^{2} \)
97 \( 1 + 96iT - 9.40e3T^{2} \)
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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

−9.759033963407493371680814424990, −9.258791489676021422699024707956, −8.765731181013702469289623122666, −7.00708310335353663380909567075, −6.70298337785274595279946965582, −5.64116369376481590908171066301, −4.68338490010821987722288262438, −3.26102440238960228639834719008, −2.35373810222280567446645419107, −0.67885384643183087730821964701, 1.32784953177614942023012462409, 2.66729083627253663831910019260, 3.68887956567564291306728119752, 5.13906495716861044519964357166, 5.89883830686684885655164319070, 6.70618412452262624674516652481, 7.74137133059656551557014749892, 8.710788321717757902828478707445, 9.811845257553129603229544174090, 10.13258366603883659783974522507

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