Properties

Label 2-1520-5.4-c1-0-29
Degree 22
Conductor 15201520
Sign 0.309+0.950i0.309 + 0.950i
Analytic cond. 12.137212.1372
Root an. cond. 3.483853.48385
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  − 2.84i·3-s + (0.691 + 2.12i)5-s + 0.145i·7-s − 5.10·9-s + 5.71·11-s − 5.24i·13-s + (6.05 − 1.96i)15-s + 7.15i·17-s + 19-s + 0.412·21-s − 0.622i·23-s + (−4.04 + 2.94i)25-s + 6.00i·27-s + 5.46·29-s + 3.77·31-s + ⋯
L(s)  = 1  − 1.64i·3-s + (0.309 + 0.950i)5-s + 0.0548i·7-s − 1.70·9-s + 1.72·11-s − 1.45i·13-s + (1.56 − 0.508i)15-s + 1.73i·17-s + 0.229·19-s + 0.0901·21-s − 0.129i·23-s + (−0.808 + 0.588i)25-s + 1.15i·27-s + 1.01·29-s + 0.678·31-s + ⋯

Functional equation

Λ(s)=(1520s/2ΓC(s)L(s)=((0.309+0.950i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1520s/2ΓC(s+1/2)L(s)=((0.309+0.950i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15201520    =    245192^{4} \cdot 5 \cdot 19
Sign: 0.309+0.950i0.309 + 0.950i
Analytic conductor: 12.137212.1372
Root analytic conductor: 3.483853.48385
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1520(609,)\chi_{1520} (609, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1520, ( :1/2), 0.309+0.950i)(2,\ 1520,\ (\ :1/2),\ 0.309 + 0.950i)

Particular Values

L(1)L(1) \approx 1.9623739121.962373912
L(12)L(\frac12) \approx 1.9623739121.962373912
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+(0.6912.12i)T 1 + (-0.691 - 2.12i)T
19 1T 1 - T
good3 1+2.84iT3T2 1 + 2.84iT - 3T^{2}
7 10.145iT7T2 1 - 0.145iT - 7T^{2}
11 15.71T+11T2 1 - 5.71T + 11T^{2}
13 1+5.24iT13T2 1 + 5.24iT - 13T^{2}
17 17.15iT17T2 1 - 7.15iT - 17T^{2}
23 1+0.622iT23T2 1 + 0.622iT - 23T^{2}
29 15.46T+29T2 1 - 5.46T + 29T^{2}
31 13.77T+31T2 1 - 3.77T + 31T^{2}
37 1+5.03iT37T2 1 + 5.03iT - 37T^{2}
41 15.77T+41T2 1 - 5.77T + 41T^{2}
43 1+3.32iT43T2 1 + 3.32iT - 43T^{2}
47 1+5.85iT47T2 1 + 5.85iT - 47T^{2}
53 16.97iT53T2 1 - 6.97iT - 53T^{2}
59 19.09T+59T2 1 - 9.09T + 59T^{2}
61 1+8.16T+61T2 1 + 8.16T + 61T^{2}
67 1+13.6iT67T2 1 + 13.6iT - 67T^{2}
71 1+2.41T+71T2 1 + 2.41T + 71T^{2}
73 1+7.44iT73T2 1 + 7.44iT - 73T^{2}
79 1+9.69T+79T2 1 + 9.69T + 79T^{2}
83 1+2.17iT83T2 1 + 2.17iT - 83T^{2}
89 1+3.90T+89T2 1 + 3.90T + 89T^{2}
97 14.98iT97T2 1 - 4.98iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.116874323884118177732056541440, −8.287086326222519113161816348626, −7.60760274764492804996292881954, −6.81132953531931430246541974082, −6.20654788642185527333516964842, −5.72846058891521949810104398215, −3.96817626991333269538831862234, −2.98443584051759968539176604089, −1.96345648673209389123102334946, −0.975240058292870222364100414435, 1.18480890710461835710753949727, 2.79602950245416530076975536493, 4.11978257236709199208711908587, 4.39913364856039961940055294990, 5.20191899164261512619694374484, 6.22497232793593668368454619660, 7.09949181505727591634125011624, 8.587509997347669803833019339851, 8.994478917215631643926262168220, 9.688145478827945684007999974394

Graph of the ZZ-function along the critical line