Properties

Label 2-720-80.27-c1-0-44
Degree $2$
Conductor $720$
Sign $0.454 + 0.890i$
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  + (0.516 + 1.31i)2-s + (−1.46 + 1.36i)4-s + (−2.07 + 0.841i)5-s + (−1.13 − 1.13i)7-s + (−2.54 − 1.22i)8-s + (−2.17 − 2.29i)10-s + (2.32 − 2.32i)11-s − 1.36i·13-s + (0.911 − 2.08i)14-s + (0.297 − 3.98i)16-s + (−5.25 − 5.25i)17-s + (−3.69 + 3.69i)19-s + (1.89 − 4.05i)20-s + (4.25 + 1.85i)22-s + (0.911 − 0.911i)23-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)2-s + (−0.732 + 0.680i)4-s + (−0.926 + 0.376i)5-s + (−0.430 − 0.430i)7-s + (−0.901 − 0.433i)8-s + (−0.688 − 0.724i)10-s + (0.700 − 0.700i)11-s − 0.378i·13-s + (0.243 − 0.558i)14-s + (0.0744 − 0.997i)16-s + (−1.27 − 1.27i)17-s + (−0.848 + 0.848i)19-s + (0.422 − 0.906i)20-s + (0.907 + 0.395i)22-s + (0.189 − 0.189i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.454 + 0.890i$
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: \((2,\ 720,\ (\ :1/2),\ 0.454 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433983 - 0.265619i\)
\(L(\frac12)\) \(\approx\) \(0.433983 - 0.265619i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.516 - 1.31i)T \)
3 \( 1 \)
5 \( 1 + (2.07 - 0.841i)T \)
good7 \( 1 + (1.13 + 1.13i)T + 7iT^{2} \)
11 \( 1 + (-2.32 + 2.32i)T - 11iT^{2} \)
13 \( 1 + 1.36iT - 13T^{2} \)
17 \( 1 + (5.25 + 5.25i)T + 17iT^{2} \)
19 \( 1 + (3.69 - 3.69i)T - 19iT^{2} \)
23 \( 1 + (-0.911 + 0.911i)T - 23iT^{2} \)
29 \( 1 + (-2.37 - 2.37i)T + 29iT^{2} \)
31 \( 1 - 0.242iT - 31T^{2} \)
37 \( 1 + 3.34iT - 37T^{2} \)
41 \( 1 + 2.66iT - 41T^{2} \)
43 \( 1 + 9.04iT - 43T^{2} \)
47 \( 1 + (7.87 - 7.87i)T - 47iT^{2} \)
53 \( 1 - 5.80T + 53T^{2} \)
59 \( 1 + (5.91 + 5.91i)T + 59iT^{2} \)
61 \( 1 + (6.67 - 6.67i)T - 61iT^{2} \)
67 \( 1 + 4.54iT - 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 + (1.49 + 1.49i)T + 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 3.26T + 83T^{2} \)
89 \( 1 + 9.77T + 89T^{2} \)
97 \( 1 + (1.63 + 1.63i)T + 97iT^{2} \)
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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

−10.26617049927382666203568330597, −9.023680798232490029877963716190, −8.482046858627582436480965669565, −7.41397234792562738047738921748, −6.81621647900449118778113854449, −6.01789317718488279383152268657, −4.69391745286133140685625429313, −3.88855834031085107050435788654, −2.98884959021183300916978903378, −0.23470889926297489077378308303, 1.65934967960976533025809081225, 2.96901056837920697978508107191, 4.26709790734296737981614829992, 4.55082227099745991807945890616, 6.08029248057805756833385692900, 6.88596332340198038004466779565, 8.371462896310872926471288182533, 8.913060677305811249268406630251, 9.735872869219360795552413828692, 10.78017347323333377078941627734

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