Properties

Label 2-3660-3660.1499-c0-0-0
Degree 22
Conductor 36603660
Sign 0.880+0.473i-0.880 + 0.473i
Analytic cond. 1.826571.82657
Root an. cond. 1.351501.35150
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.309 + 0.951i)3-s + (−0.104 − 0.994i)4-s + (−0.994 − 0.104i)5-s + (−0.499 − 0.866i)6-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.743 − 0.669i)10-s + (0.978 + 0.207i)12-s + (0.406 − 0.913i)15-s + (−0.978 + 0.207i)16-s + (−0.692 + 1.80i)17-s + (0.978 − 0.207i)18-s + (0.278 + 0.309i)19-s + i·20-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.309 + 0.951i)3-s + (−0.104 − 0.994i)4-s + (−0.994 − 0.104i)5-s + (−0.499 − 0.866i)6-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.743 − 0.669i)10-s + (0.978 + 0.207i)12-s + (0.406 − 0.913i)15-s + (−0.978 + 0.207i)16-s + (−0.692 + 1.80i)17-s + (0.978 − 0.207i)18-s + (0.278 + 0.309i)19-s + i·20-s + ⋯

Functional equation

Λ(s)=(3660s/2ΓC(s)L(s)=((0.880+0.473i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3660s/2ΓC(s)L(s)=((0.880+0.473i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36603660    =    2235612^{2} \cdot 3 \cdot 5 \cdot 61
Sign: 0.880+0.473i-0.880 + 0.473i
Analytic conductor: 1.826571.82657
Root analytic conductor: 1.351501.35150
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3660(1499,)\chi_{3660} (1499, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3660, ( :0), 0.880+0.473i)(2,\ 3660,\ (\ :0),\ -0.880 + 0.473i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.34093511220.3409351122
L(12)L(\frac12) \approx 0.34093511220.3409351122
L(1)L(1) 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+(0.6690.743i)T 1 + (0.669 - 0.743i)T
3 1+(0.3090.951i)T 1 + (0.309 - 0.951i)T
5 1+(0.994+0.104i)T 1 + (0.994 + 0.104i)T
61 1+(0.406+0.913i)T 1 + (0.406 + 0.913i)T
good7 1+(0.9940.104i)T2 1 + (0.994 - 0.104i)T^{2}
11 1iT2 1 - iT^{2}
13 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1+(0.6921.80i)T+(0.7430.669i)T2 1 + (0.692 - 1.80i)T + (-0.743 - 0.669i)T^{2}
19 1+(0.2780.309i)T+(0.104+0.994i)T2 1 + (-0.278 - 0.309i)T + (-0.104 + 0.994i)T^{2}
23 1+(1.65+0.262i)T+(0.9510.309i)T2 1 + (-1.65 + 0.262i)T + (0.951 - 0.309i)T^{2}
29 1+(0.8660.5i)T2 1 + (0.866 - 0.5i)T^{2}
31 1+(0.6851.05i)T+(0.406+0.913i)T2 1 + (-0.685 - 1.05i)T + (-0.406 + 0.913i)T^{2}
37 1+(0.5870.809i)T2 1 + (-0.587 - 0.809i)T^{2}
41 1+(0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
43 1+(0.7430.669i)T2 1 + (0.743 - 0.669i)T^{2}
47 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
53 1+(1.84+0.292i)T+(0.951+0.309i)T2 1 + (1.84 + 0.292i)T + (0.951 + 0.309i)T^{2}
59 1+(0.4060.913i)T2 1 + (-0.406 - 0.913i)T^{2}
67 1+(0.207+0.978i)T2 1 + (-0.207 + 0.978i)T^{2}
71 1+(0.207+0.978i)T2 1 + (0.207 + 0.978i)T^{2}
73 1+(0.9780.207i)T2 1 + (0.978 - 0.207i)T^{2}
79 1+(1.860.715i)T+(0.7430.669i)T2 1 + (1.86 - 0.715i)T + (0.743 - 0.669i)T^{2}
83 1+(0.4131.94i)T+(0.913+0.406i)T2 1 + (-0.413 - 1.94i)T + (-0.913 + 0.406i)T^{2}
89 1+(0.5870.809i)T2 1 + (0.587 - 0.809i)T^{2}
97 1+(0.913+0.406i)T2 1 + (0.913 + 0.406i)T^{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

−8.987182642751449053505569908335, −8.383788397578421986479561391089, −7.951831777308973066696511015687, −6.75273126037143117693904434080, −6.42162960347068694172849721479, −5.29328990245356391732924262436, −4.74531633995892321812693304287, −3.96401774883402352644709289494, −3.03512931337064082545594591630, −1.34600873716073931441511683207, 0.29298030986127345739884771779, 1.37394029671026281640817388332, 2.76570477647167372070340684524, 3.06590545734824126885768566291, 4.47953758195477566558072649696, 5.01891149735695364660395637676, 6.42834913700042059532181949502, 7.12894884136875445192092234452, 7.53932689314145538570568423444, 8.231474357195028088817389313281

Graph of the ZZ-function along the critical line