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Properties

Label 2-720-80.29-c1-0-5
Degree $2$
Conductor $720$
Sign $-0.849 + 0.528i$
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  + (0.750 + 1.19i)2-s + (−0.874 + 1.79i)4-s + (−1.07 + 1.95i)5-s − 1.22·7-s + (−2.81 + 0.302i)8-s + (−3.15 + 0.178i)10-s + (−1.38 + 1.38i)11-s + (2.12 − 2.12i)13-s + (−0.916 − 1.46i)14-s + (−2.47 − 3.14i)16-s + 6.00i·17-s + (−3.06 − 3.06i)19-s + (−2.58 − 3.65i)20-s + (−2.69 − 0.619i)22-s − 2.90·23-s + ⋯
L(s)  = 1  + (0.530 + 0.847i)2-s + (−0.437 + 0.899i)4-s + (−0.481 + 0.876i)5-s − 0.461·7-s + (−0.994 + 0.106i)8-s + (−0.998 + 0.0565i)10-s + (−0.416 + 0.416i)11-s + (0.588 − 0.588i)13-s + (−0.244 − 0.391i)14-s + (−0.618 − 0.786i)16-s + 1.45i·17-s + (−0.702 − 0.702i)19-s + (−0.577 − 0.816i)20-s + (−0.574 − 0.132i)22-s − 0.606·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.849 + 0.528i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.849 + 0.528i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.244861 - 0.857573i\)
\(L(\frac12)\) \(\approx\) \(0.244861 - 0.857573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.750 - 1.19i)T \)
3 \( 1 \)
5 \( 1 + (1.07 - 1.95i)T \)
good7 \( 1 + 1.22T + 7T^{2} \)
11 \( 1 + (1.38 - 1.38i)T - 11iT^{2} \)
13 \( 1 + (-2.12 + 2.12i)T - 13iT^{2} \)
17 \( 1 - 6.00iT - 17T^{2} \)
19 \( 1 + (3.06 + 3.06i)T + 19iT^{2} \)
23 \( 1 + 2.90T + 23T^{2} \)
29 \( 1 + (3.18 + 3.18i)T + 29iT^{2} \)
31 \( 1 + 3.88T + 31T^{2} \)
37 \( 1 + (-2.44 - 2.44i)T + 37iT^{2} \)
41 \( 1 - 2.38iT - 41T^{2} \)
43 \( 1 + (-9.00 - 9.00i)T + 43iT^{2} \)
47 \( 1 - 0.586iT - 47T^{2} \)
53 \( 1 + (-2.36 - 2.36i)T + 53iT^{2} \)
59 \( 1 + (8.43 - 8.43i)T - 59iT^{2} \)
61 \( 1 + (-9.98 - 9.98i)T + 61iT^{2} \)
67 \( 1 + (3.82 - 3.82i)T - 67iT^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 - 1.31T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + (-2.91 + 2.91i)T - 83iT^{2} \)
89 \( 1 - 9.58iT - 89T^{2} \)
97 \( 1 - 9.45iT - 97T^{2} \)
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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

−10.90345885846796250703668654997, −10.12888966917850665021196434152, −8.970589245580943821160764024413, −8.006239830482199346464102130262, −7.45336967704280854315975422277, −6.33689451950451224071250390338, −5.92206663922050400095771278748, −4.44103313761794952034641497520, −3.65778967417546125712416629399, −2.59184618112804488074151507408, 0.37383087944712621353486974065, 1.95024857188398519250226942901, 3.38609937358278156314724185140, 4.18193347259660926620894614043, 5.21474164390484698879519889144, 6.01143360115785250721187011261, 7.26309108374772066041400336360, 8.490514974716141921368748112107, 9.163775585657611508668770279879, 9.928701970552063464421815639732

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