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Properties

Label 2-720-80.27-c1-0-54
Degree $2$
Conductor $720$
Sign $-0.168 + 0.985i$
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  + (0.897 + 1.09i)2-s + (−0.390 + 1.96i)4-s + (−0.140 − 2.23i)5-s + (−2.83 − 2.83i)7-s + (−2.49 + 1.33i)8-s + (2.31 − 2.15i)10-s + (−4.36 + 4.36i)11-s − 4.99i·13-s + (0.555 − 5.63i)14-s + (−3.69 − 1.53i)16-s + (−2.27 − 2.27i)17-s + (1.45 − 1.45i)19-s + (4.43 + 0.596i)20-s + (−8.67 − 0.856i)22-s + (−1.28 + 1.28i)23-s + ⋯
L(s)  = 1  + (0.634 + 0.773i)2-s + (−0.195 + 0.980i)4-s + (−0.0627 − 0.998i)5-s + (−1.07 − 1.07i)7-s + (−0.882 + 0.470i)8-s + (0.731 − 0.681i)10-s + (−1.31 + 1.31i)11-s − 1.38i·13-s + (0.148 − 1.50i)14-s + (−0.923 − 0.383i)16-s + (−0.551 − 0.551i)17-s + (0.334 − 0.334i)19-s + (0.991 + 0.133i)20-s + (−1.85 − 0.182i)22-s + (−0.268 + 0.268i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.168 + 0.985i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.168 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.426944 - 0.506264i\)
\(L(\frac12)\) \(\approx\) \(0.426944 - 0.506264i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.897 - 1.09i)T \)
3 \( 1 \)
5 \( 1 + (0.140 + 2.23i)T \)
good7 \( 1 + (2.83 + 2.83i)T + 7iT^{2} \)
11 \( 1 + (4.36 - 4.36i)T - 11iT^{2} \)
13 \( 1 + 4.99iT - 13T^{2} \)
17 \( 1 + (2.27 + 2.27i)T + 17iT^{2} \)
19 \( 1 + (-1.45 + 1.45i)T - 19iT^{2} \)
23 \( 1 + (1.28 - 1.28i)T - 23iT^{2} \)
29 \( 1 + (-0.965 - 0.965i)T + 29iT^{2} \)
31 \( 1 + 0.703iT - 31T^{2} \)
37 \( 1 + 6.29iT - 37T^{2} \)
41 \( 1 - 0.772iT - 41T^{2} \)
43 \( 1 - 4.84iT - 43T^{2} \)
47 \( 1 + (-0.450 + 0.450i)T - 47iT^{2} \)
53 \( 1 + 4.17T + 53T^{2} \)
59 \( 1 + (-2.23 - 2.23i)T + 59iT^{2} \)
61 \( 1 + (-0.794 + 0.794i)T - 61iT^{2} \)
67 \( 1 + 13.2iT - 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + (-10.3 - 10.3i)T + 73iT^{2} \)
79 \( 1 + 4.49T + 79T^{2} \)
83 \( 1 - 7.59T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + (0.751 + 0.751i)T + 97iT^{2} \)
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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

−9.963471790532837449080548110373, −9.358353855963936370461675309897, −8.053359698099472328737340995542, −7.56064533816766975352353819147, −6.72052696365962051101176878535, −5.48701243275722868710901585654, −4.84402413254057702882768474404, −3.86829508637402562437163850747, −2.69696957040700046378839960711, −0.25546787701102881769671309922, 2.24544016272569021538691050718, 2.97414017853172460710960631636, 3.86216135568916364274329947298, 5.32219310119929293912469301966, 6.16526052731538738345551543923, 6.69016449247968823748826009093, 8.252853041652164388034025866106, 9.170430524068399736767029961240, 10.01187475810299220408247202656, 10.75126527194594579588979486534

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