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Properties

Label 2-720-80.69-c1-0-13
Degree $2$
Conductor $720$
Sign $-0.906 - 0.423i$
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  + (0.550 + 1.30i)2-s + (−1.39 + 1.43i)4-s + (−0.162 + 2.23i)5-s + 2.93·7-s + (−2.63 − 1.02i)8-s + (−2.99 + 1.01i)10-s + (0.663 + 0.663i)11-s + (1.12 + 1.12i)13-s + (1.61 + 3.82i)14-s + (−0.112 − 3.99i)16-s + 7.47i·17-s + (0.423 − 0.423i)19-s + (−2.97 − 3.34i)20-s + (−0.499 + 1.23i)22-s − 6.17·23-s + ⋯
L(s)  = 1  + (0.389 + 0.921i)2-s + (−0.697 + 0.716i)4-s + (−0.0724 + 0.997i)5-s + 1.10·7-s + (−0.931 − 0.363i)8-s + (−0.946 + 0.321i)10-s + (0.200 + 0.200i)11-s + (0.312 + 0.312i)13-s + (0.431 + 1.02i)14-s + (−0.0281 − 0.999i)16-s + 1.81i·17-s + (0.0971 − 0.0971i)19-s + (−0.664 − 0.747i)20-s + (−0.106 + 0.262i)22-s − 1.28·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.906 - 0.423i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.906 - 0.423i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.372807 + 1.67877i\)
\(L(\frac12)\) \(\approx\) \(0.372807 + 1.67877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.550 - 1.30i)T \)
3 \( 1 \)
5 \( 1 + (0.162 - 2.23i)T \)
good7 \( 1 - 2.93T + 7T^{2} \)
11 \( 1 + (-0.663 - 0.663i)T + 11iT^{2} \)
13 \( 1 + (-1.12 - 1.12i)T + 13iT^{2} \)
17 \( 1 - 7.47iT - 17T^{2} \)
19 \( 1 + (-0.423 + 0.423i)T - 19iT^{2} \)
23 \( 1 + 6.17T + 23T^{2} \)
29 \( 1 + (-2.95 + 2.95i)T - 29iT^{2} \)
31 \( 1 - 1.82T + 31T^{2} \)
37 \( 1 + (5.53 - 5.53i)T - 37iT^{2} \)
41 \( 1 + 12.3iT - 41T^{2} \)
43 \( 1 + (-0.897 + 0.897i)T - 43iT^{2} \)
47 \( 1 + 4.12iT - 47T^{2} \)
53 \( 1 + (0.146 - 0.146i)T - 53iT^{2} \)
59 \( 1 + (-7.72 - 7.72i)T + 59iT^{2} \)
61 \( 1 + (7.37 - 7.37i)T - 61iT^{2} \)
67 \( 1 + (-8.68 - 8.68i)T + 67iT^{2} \)
71 \( 1 + 8.95iT - 71T^{2} \)
73 \( 1 - 0.174T + 73T^{2} \)
79 \( 1 - 3.06T + 79T^{2} \)
83 \( 1 + (-9.18 - 9.18i)T + 83iT^{2} \)
89 \( 1 - 8.71iT - 89T^{2} \)
97 \( 1 + 10.5iT - 97T^{2} \)
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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

−10.71420026960731184458526000116, −9.996625326968417406438445184903, −8.626207530084492832139340026367, −8.130395247701249253269784234037, −7.22693902176541669221505561577, −6.36693922636923870551475095044, −5.61180313564368281691961056701, −4.33937781010138201415252589293, −3.65033088562178661603489614886, −2.04367988285871888897768744378, 0.832385091141064246515719668877, 2.01340711675796226887478849197, 3.43867329677396487738258603303, 4.67917875228106388580976152165, 5.05424527821575975716698106134, 6.15739896668122556839314750971, 7.73829099637066106504253889891, 8.478901895496292440948386361497, 9.302123168309388342921067518471, 10.06140718246108506957423596164

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