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Properties

Label 2-720-80.27-c1-0-30
Degree $2$
Conductor $720$
Sign $0.669 - 0.743i$
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  + (1.29 + 0.558i)2-s + (1.37 + 1.45i)4-s + (−1.49 − 1.66i)5-s + (2.40 + 2.40i)7-s + (0.977 + 2.65i)8-s + (−1.00 − 2.99i)10-s + (2.67 − 2.67i)11-s + 2.40i·13-s + (1.78 + 4.46i)14-s + (−0.212 + 3.99i)16-s + (0.0750 + 0.0750i)17-s + (2.67 − 2.67i)19-s + (0.366 − 4.45i)20-s + (4.97 − 1.98i)22-s + (−2.12 + 2.12i)23-s + ⋯
L(s)  = 1  + (0.918 + 0.394i)2-s + (0.688 + 0.725i)4-s + (−0.666 − 0.745i)5-s + (0.908 + 0.908i)7-s + (0.345 + 0.938i)8-s + (−0.318 − 0.947i)10-s + (0.807 − 0.807i)11-s + 0.666i·13-s + (0.475 + 1.19i)14-s + (−0.0532 + 0.998i)16-s + (0.0182 + 0.0182i)17-s + (0.613 − 0.613i)19-s + (0.0819 − 0.996i)20-s + (1.06 − 0.422i)22-s + (−0.442 + 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.669 - 0.743i$
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: \((2,\ 720,\ (\ :1/2),\ 0.669 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46883 + 1.09891i\)
\(L(\frac12)\) \(\approx\) \(2.46883 + 1.09891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.558i)T \)
3 \( 1 \)
5 \( 1 + (1.49 + 1.66i)T \)
good7 \( 1 + (-2.40 - 2.40i)T + 7iT^{2} \)
11 \( 1 + (-2.67 + 2.67i)T - 11iT^{2} \)
13 \( 1 - 2.40iT - 13T^{2} \)
17 \( 1 + (-0.0750 - 0.0750i)T + 17iT^{2} \)
19 \( 1 + (-2.67 + 2.67i)T - 19iT^{2} \)
23 \( 1 + (2.12 - 2.12i)T - 23iT^{2} \)
29 \( 1 + (-3.95 - 3.95i)T + 29iT^{2} \)
31 \( 1 + 1.65iT - 31T^{2} \)
37 \( 1 + 2.53iT - 37T^{2} \)
41 \( 1 + 1.70iT - 41T^{2} \)
43 \( 1 - 3.84iT - 43T^{2} \)
47 \( 1 + (2.15 - 2.15i)T - 47iT^{2} \)
53 \( 1 - 1.29T + 53T^{2} \)
59 \( 1 + (5.29 + 5.29i)T + 59iT^{2} \)
61 \( 1 + (-10.2 + 10.2i)T - 61iT^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 + 2.27T + 71T^{2} \)
73 \( 1 + (9.99 + 9.99i)T + 73iT^{2} \)
79 \( 1 + 8.70T + 79T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (-5.00 - 5.00i)T + 97iT^{2} \)
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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

−11.10041346284430397727247403010, −9.325234738836912446795771010420, −8.599842130042365891964013326836, −7.969185106562964494720254234357, −6.93781198447687009150976006845, −5.86735895547144286810326899160, −5.04984976308409473225744386998, −4.26687283189707826998828532787, −3.18082314061217715042505224458, −1.65442082248678146133020744321, 1.29023126328936466492519661417, 2.75651006465926179156958068860, 3.97009867926922923646335928456, 4.44758177841187738908287099710, 5.70382313517568290570161276302, 6.86145273378143906076888106370, 7.40513056672985201674665896580, 8.351769724586579373413082575533, 10.08213010353129870325562386623, 10.27197597816365246098577224992

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