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Properties

Label 2-720-45.34-c1-0-21
Degree $2$
Conductor $720$
Sign $0.999 + 0.00269i$
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  + (−0.690 + 1.58i)3-s + (1.99 − 1.00i)5-s + (1.11 + 0.644i)7-s + (−2.04 − 2.19i)9-s + (2.54 − 4.41i)11-s + (3.09 − 1.78i)13-s + (0.225 + 3.86i)15-s − 0.895i·17-s − 5.34·19-s + (−1.79 + 1.32i)21-s + (4.38 − 2.53i)23-s + (2.96 − 4.02i)25-s + (4.89 − 1.73i)27-s + (−1.5 + 2.59i)29-s + (−3.29 − 5.71i)31-s + ⋯
L(s)  = 1  + (−0.398 + 0.917i)3-s + (0.892 − 0.451i)5-s + (0.421 + 0.243i)7-s + (−0.682 − 0.731i)9-s + (0.767 − 1.32i)11-s + (0.857 − 0.495i)13-s + (0.0582 + 0.998i)15-s − 0.217i·17-s − 1.22·19-s + (−0.391 + 0.289i)21-s + (0.914 − 0.528i)23-s + (0.592 − 0.805i)25-s + (0.942 − 0.334i)27-s + (−0.278 + 0.482i)29-s + (−0.592 − 1.02i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.999 + 0.00269i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.999 + 0.00269i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66214 - 0.00223641i\)
\(L(\frac12)\) \(\approx\) \(1.66214 - 0.00223641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.690 - 1.58i)T \)
5 \( 1 + (-1.99 + 1.00i)T \)
good7 \( 1 + (-1.11 - 0.644i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.54 + 4.41i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.09 + 1.78i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.895iT - 17T^{2} \)
19 \( 1 + 5.34T + 19T^{2} \)
23 \( 1 + (-4.38 + 2.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.29 + 5.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.24iT - 37T^{2} \)
41 \( 1 + (-3.92 - 6.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.46 - 5.46i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.57 + 1.48i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.78iT - 53T^{2} \)
59 \( 1 + (-2.87 - 4.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.17 - 3.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.39 - 4.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.34T + 71T^{2} \)
73 \( 1 + 9.34iT - 73T^{2} \)
79 \( 1 + (-0.370 + 0.642i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.97 - 4.60i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.24T + 89T^{2} \)
97 \( 1 + (2.99 + 1.72i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.60288156660340871400052828521, −9.409499135803636294353086448481, −8.882280454155641348565540672193, −8.215554280954697575376804813020, −6.38027138103629406251118231237, −5.97128872463973113769005550388, −5.02789341729405913181015122514, −4.04470122441978125938517679346, −2.84220814642831326813051124062, −1.06003824912197139798975400825, 1.49584832387724757488527027396, 2.22785596311731643663686944464, 3.95253112106624186099734866204, 5.17296551637279303611454337854, 6.17432817085334799244763027898, 6.85362400463265829143770530992, 7.51420063041616895568229391148, 8.778533194788781528604197157783, 9.458689460983131189282003172811, 10.82331855264020598605257325809

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