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Properties

Label 2-720-15.14-c2-0-21
Degree 22
Conductor 720720
Sign −0.842+0.539i-0.842 + 0.539i
Analytic cond. 19.618519.6185
Root an. cond. 4.429284.42928
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.229 − 4.99i)5-s − 8.73i·7-s − 10.4i·11-s + 5.38i·13-s + 26.2·17-s + 2.70·19-s − 33.2·23-s + (−24.8 − 2.28i)25-s − 17.4i·29-s − 48.3·31-s + (−43.6 − 2.00i)35-s + 66.2i·37-s − 14.7i·41-s − 28.4i·43-s + 35.9·47-s + ⋯
L(s)  = 1  + (0.0458 − 0.998i)5-s − 1.24i·7-s − 0.953i·11-s + 0.414i·13-s + 1.54·17-s + 0.142·19-s − 1.44·23-s + (−0.995 − 0.0915i)25-s − 0.602i·29-s − 1.56·31-s + (−1.24 − 0.0572i)35-s + 1.79i·37-s − 0.359i·41-s − 0.661i·43-s + 0.764·47-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((−0.842+0.539i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 + 0.539i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1) L(s)=((−0.842+0.539i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.842 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: −0.842+0.539i-0.842 + 0.539i
Analytic conductor: 19.618519.6185
Root analytic conductor: 4.429284.42928
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ720(449,⋅)\chi_{720} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1), −0.842+0.539i)(2,\ 720,\ (\ :1),\ -0.842 + 0.539i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.3491398551.349139855
L(12)L(\frac12) ≈\approx 1.3491398551.349139855
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
5 1+(−0.229+4.99i)T 1 + (-0.229 + 4.99i)T
good7 1+8.73iT−49T2 1 + 8.73iT - 49T^{2}
11 1+10.4iT−121T2 1 + 10.4iT - 121T^{2}
13 1−5.38iT−169T2 1 - 5.38iT - 169T^{2}
17 1−26.2T+289T2 1 - 26.2T + 289T^{2}
19 1−2.70T+361T2 1 - 2.70T + 361T^{2}
23 1+33.2T+529T2 1 + 33.2T + 529T^{2}
29 1+17.4iT−841T2 1 + 17.4iT - 841T^{2}
31 1+48.3T+961T2 1 + 48.3T + 961T^{2}
37 1−66.2iT−1.36e3T2 1 - 66.2iT - 1.36e3T^{2}
41 1+14.7iT−1.68e3T2 1 + 14.7iT - 1.68e3T^{2}
43 1+28.4iT−1.84e3T2 1 + 28.4iT - 1.84e3T^{2}
47 1−35.9T+2.20e3T2 1 - 35.9T + 2.20e3T^{2}
53 1−42.2T+2.80e3T2 1 - 42.2T + 2.80e3T^{2}
59 1+55.9iT−3.48e3T2 1 + 55.9iT - 3.48e3T^{2}
61 1+96.1T+3.72e3T2 1 + 96.1T + 3.72e3T^{2}
67 1+15.4iT−4.48e3T2 1 + 15.4iT - 4.48e3T^{2}
71 1+13.5iT−5.04e3T2 1 + 13.5iT - 5.04e3T^{2}
73 1+63.7iT−5.32e3T2 1 + 63.7iT - 5.32e3T^{2}
79 1+94.5T+6.24e3T2 1 + 94.5T + 6.24e3T^{2}
83 1−19.4T+6.88e3T2 1 - 19.4T + 6.88e3T^{2}
89 1+118.iT−7.92e3T2 1 + 118. iT - 7.92e3T^{2}
97 1−100.iT−9.40e3T2 1 - 100. iT - 9.40e3T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−9.918831248200409166756845426911, −9.000807142049985870522989895856, −8.042488365131365908359540233303, −7.47121094034183205293075019506, −6.17529501451856531313937925755, −5.35107025224837554037339712840, −4.21026928095079221022445614426, −3.45900661445928541514594057888, −1.57997579684810745318675853365, −0.47344445501328804642609137115, 1.89208143120996826081844911597, 2.86381937397530079513659924042, 3.94755686852305707710969228853, 5.49926100248261742318899445620, 5.90849085905738720967197077539, 7.22712795755972166732466208401, 7.77394220690894412262773216118, 8.956219903761548108715251149511, 9.793385423675127051846125627866, 10.43597445369790580321917725552

Graph of the ZZ-function along the critical line