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Properties

Label 2-720-16.13-c1-0-16
Degree $2$
Conductor $720$
Sign $0.657 - 0.753i$
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  + (−0.114 + 1.40i)2-s + (−1.97 − 0.323i)4-s + (−0.707 + 0.707i)5-s − 0.690i·7-s + (0.681 − 2.74i)8-s + (−0.915 − 1.07i)10-s + (3.06 − 3.06i)11-s + (−2.33 − 2.33i)13-s + (0.973 + 0.0791i)14-s + (3.79 + 1.27i)16-s + 5.28·17-s + (5.38 + 5.38i)19-s + (1.62 − 1.16i)20-s + (3.96 + 4.66i)22-s + 1.60i·23-s + ⋯
L(s)  = 1  + (−0.0810 + 0.996i)2-s + (−0.986 − 0.161i)4-s + (−0.316 + 0.316i)5-s − 0.261i·7-s + (0.241 − 0.970i)8-s + (−0.289 − 0.340i)10-s + (0.922 − 0.922i)11-s + (−0.648 − 0.648i)13-s + (0.260 + 0.0211i)14-s + (0.947 + 0.318i)16-s + 1.28·17-s + (1.23 + 1.23i)19-s + (0.363 − 0.260i)20-s + (0.844 + 0.994i)22-s + 0.335i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18053 + 0.536776i\)
\(L(\frac12)\) \(\approx\) \(1.18053 + 0.536776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.114 - 1.40i)T \)
3 \( 1 \)
5 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + 0.690iT - 7T^{2} \)
11 \( 1 + (-3.06 + 3.06i)T - 11iT^{2} \)
13 \( 1 + (2.33 + 2.33i)T + 13iT^{2} \)
17 \( 1 - 5.28T + 17T^{2} \)
19 \( 1 + (-5.38 - 5.38i)T + 19iT^{2} \)
23 \( 1 - 1.60iT - 23T^{2} \)
29 \( 1 + (1.70 + 1.70i)T + 29iT^{2} \)
31 \( 1 + 4.69T + 31T^{2} \)
37 \( 1 + (-7.89 + 7.89i)T - 37iT^{2} \)
41 \( 1 - 5.49iT - 41T^{2} \)
43 \( 1 + (0.256 - 0.256i)T - 43iT^{2} \)
47 \( 1 - 4.60T + 47T^{2} \)
53 \( 1 + (-4.99 + 4.99i)T - 53iT^{2} \)
59 \( 1 + (1.46 - 1.46i)T - 59iT^{2} \)
61 \( 1 + (-9.33 - 9.33i)T + 61iT^{2} \)
67 \( 1 + (1.94 + 1.94i)T + 67iT^{2} \)
71 \( 1 - 2.32iT - 71T^{2} \)
73 \( 1 + 1.29iT - 73T^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + (7.30 + 7.30i)T + 83iT^{2} \)
89 \( 1 + 1.81iT - 89T^{2} \)
97 \( 1 - 5.27T + 97T^{2} \)
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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

−10.20592761925784299896700120051, −9.644717099430251834804404298417, −8.640169709118580653652857952746, −7.63313094683937549000188564978, −7.33543241141978554924890482163, −5.92746720444489778736713710561, −5.53303925969908202930481762037, −4.03962013011029510418612980738, −3.30559112463953565733844245065, −0.952945977671787045555353798777, 1.12389123119584150306041364107, 2.46439937237722724797769457304, 3.68599292942899088021066197603, 4.63942998871088874838687584516, 5.44730914152981958652369918399, 7.00316530446784331747628004295, 7.77336887099625064928800168983, 9.016339596053570030955175387702, 9.417047435737220399505845277655, 10.20177330496334465871944833817

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