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Properties

Label 2-240-60.59-c1-0-11
Degree 22
Conductor 240240
Sign −0.912+0.408i-0.912 + 0.408i
Analytic cond. 1.916401.91640
Root an. cond. 1.384341.38434
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 1.58i)3-s − 2.23i·5-s − 4.24·7-s + (−2.00 + 2.23i)9-s + (−3.53 + 1.58i)15-s + (3 + 6.70i)21-s − 9.48i·23-s − 5.00·25-s + (4.94 + 1.58i)27-s − 8.94i·29-s + 9.48i·35-s − 4.47i·41-s + 12.7·43-s + (5.00 + 4.47i)45-s + 9.48i·47-s + ⋯
L(s)  = 1  + (−0.408 − 0.912i)3-s − 0.999i·5-s − 1.60·7-s + (−0.666 + 0.745i)9-s + (−0.912 + 0.408i)15-s + (0.654 + 1.46i)21-s − 1.97i·23-s − 1.00·25-s + (0.952 + 0.304i)27-s − 1.66i·29-s + 1.60i·35-s − 0.698i·41-s + 1.94·43-s + (0.745 + 0.666i)45-s + 1.38i·47-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=((−0.912+0.408i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+1/2) L(s)=((−0.912+0.408i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: −0.912+0.408i-0.912 + 0.408i
Analytic conductor: 1.916401.91640
Root analytic conductor: 1.384341.38434
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ240(239,⋅)\chi_{240} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 240, ( :1/2), −0.912+0.408i)(2,\ 240,\ (\ :1/2),\ -0.912 + 0.408i)

Particular Values

L(1)L(1) ≈\approx 0.133376−0.624942i0.133376 - 0.624942i
L(12)L(\frac12) ≈\approx 0.133376−0.624942i0.133376 - 0.624942i
L(32)L(\frac{3}{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+(0.707+1.58i)T 1 + (0.707 + 1.58i)T
5 1+2.23iT 1 + 2.23iT
good7 1+4.24T+7T2 1 + 4.24T + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1−13T2 1 - 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1+9.48iT−23T2 1 + 9.48iT - 23T^{2}
29 1+8.94iT−29T2 1 + 8.94iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1+4.47iT−41T2 1 + 4.47iT - 41T^{2}
43 1−12.7T+43T2 1 - 12.7T + 43T^{2}
47 1−9.48iT−47T2 1 - 9.48iT - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+8T+61T2 1 + 8T + 61T^{2}
67 1−4.24T+67T2 1 - 4.24T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1+9.48iT−83T2 1 + 9.48iT - 83T^{2}
89 1−17.8iT−89T2 1 - 17.8iT - 89T^{2}
97 1−97T2 1 - 97T^{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

−12.17364115039456872930094446837, −10.82804725876035594364390919149, −9.697860553828572234477468062393, −8.766092299948606886619060035612, −7.69509883303102088749725745509, −6.48589477155513661571079341489, −5.81924034855598533822454440195, −4.33195339279635486193965981887, −2.55873991987141084249143622113, −0.52268030501985972374441723664, 3.04161388429217758519522527978, 3.77025411668760495326681732912, 5.49883589410066484416491785194, 6.39920909926275457995919603208, 7.33515196440029765902914468121, 9.085924980307799539241083697314, 9.770053059269143864103826346517, 10.53632331909272887233558415503, 11.39441915962966927951661169798, 12.41883650815690816896739777427

Graph of the ZZ-function along the critical line