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Properties

Label 2-60-15.14-c2-0-2
Degree 22
Conductor 6060
Sign 0.929+0.368i0.929 + 0.368i
Analytic cond. 1.634881.63488
Root an. cond. 1.278621.27862
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  + (2.23 − 2i)3-s + (2.23 + 4.47i)5-s − 8i·7-s + (1.00 − 8.94i)9-s + 8.94i·11-s + 12i·13-s + (13.9 + 5.52i)15-s − 31.3·17-s − 6·19-s + (−16 − 17.8i)21-s + 4.47·23-s + (−15.0 + 20.0i)25-s + (−15.6 − 22.0i)27-s + 26.8i·29-s + 34·31-s + ⋯
L(s)  = 1  + (0.745 − 0.666i)3-s + (0.447 + 0.894i)5-s − 1.14i·7-s + (0.111 − 0.993i)9-s + 0.813i·11-s + 0.923i·13-s + (0.929 + 0.368i)15-s − 1.84·17-s − 0.315·19-s + (−0.761 − 0.851i)21-s + 0.194·23-s + (−0.600 + 0.800i)25-s + (−0.579 − 0.814i)27-s + 0.925i·29-s + 1.09·31-s + ⋯

Functional equation

Λ(s)=(60s/2 ΓC(s) L(s)=((0.929+0.368i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(60s/2 ΓC(s+1) L(s)=((0.929+0.368i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6060    =    22⋅3⋅52^{2} \cdot 3 \cdot 5
Sign: 0.929+0.368i0.929 + 0.368i
Analytic conductor: 1.634881.63488
Root analytic conductor: 1.278621.27862
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ60(29,⋅)\chi_{60} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 60, ( :1), 0.929+0.368i)(2,\ 60,\ (\ :1),\ 0.929 + 0.368i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.45178−0.277265i1.45178 - 0.277265i
L(12)L(\frac12) ≈\approx 1.45178−0.277265i1.45178 - 0.277265i
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+(−2.23+2i)T 1 + (-2.23 + 2i)T
5 1+(−2.23−4.47i)T 1 + (-2.23 - 4.47i)T
good7 1+8iT−49T2 1 + 8iT - 49T^{2}
11 1−8.94iT−121T2 1 - 8.94iT - 121T^{2}
13 1−12iT−169T2 1 - 12iT - 169T^{2}
17 1+31.3T+289T2 1 + 31.3T + 289T^{2}
19 1+6T+361T2 1 + 6T + 361T^{2}
23 1−4.47T+529T2 1 - 4.47T + 529T^{2}
29 1−26.8iT−841T2 1 - 26.8iT - 841T^{2}
31 1−34T+961T2 1 - 34T + 961T^{2}
37 1+44iT−1.36e3T2 1 + 44iT - 1.36e3T^{2}
41 1+17.8iT−1.68e3T2 1 + 17.8iT - 1.68e3T^{2}
43 1+28iT−1.84e3T2 1 + 28iT - 1.84e3T^{2}
47 1−4.47T+2.20e3T2 1 - 4.47T + 2.20e3T^{2}
53 1−40.2T+2.80e3T2 1 - 40.2T + 2.80e3T^{2}
59 1+98.3iT−3.48e3T2 1 + 98.3iT - 3.48e3T^{2}
61 1−74T+3.72e3T2 1 - 74T + 3.72e3T^{2}
67 1−92iT−4.48e3T2 1 - 92iT - 4.48e3T^{2}
71 1+53.6iT−5.04e3T2 1 + 53.6iT - 5.04e3T^{2}
73 1−56iT−5.32e3T2 1 - 56iT - 5.32e3T^{2}
79 1+78T+6.24e3T2 1 + 78T + 6.24e3T^{2}
83 1+102.T+6.88e3T2 1 + 102.T + 6.88e3T^{2}
89 1−17.8iT−7.92e3T2 1 - 17.8iT - 7.92e3T^{2}
97 1−32iT−9.40e3T2 1 - 32iT - 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

−14.47621702430565846663890279212, −13.78758732139391218205172937444, −12.87789159468169518770330368802, −11.30211096781844578266825975358, −10.12245583978878001683934168376, −8.880679047898305080751982526464, −7.18236054184151654360121377292, −6.70381367980156212657825159348, −4.08223328678377031679040300796, −2.17131723727890984097766088136, 2.58186362428924905480915729866, 4.62424247316482503301892295422, 5.94137585119197281268068358720, 8.347297233587131691794432032574, 8.850276241935608498957155139665, 10.05556224265148204549474634162, 11.48153338687204257967085112029, 12.96917461784470061224317786048, 13.68175108496696622664396584733, 15.20792972282815476015571279133

Graph of the ZZ-function along the critical line