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Properties

Label 2-3660-3660.1499-c0-0-6
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/2 ΓC(s) L(s)=((−0.880+0.473i) Λ‾(1−s)\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/2 ΓC(s) L(s)=((−0.880+0.473i) Λ‾(1−s)\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    =    22⋅3⋅5⋅612^{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 2.2252674792.225267479
L(12)L(\frac12) ≈\approx 2.2252674792.225267479
L(1)L(1) 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+(−0.669+0.743i)T 1 + (-0.669 + 0.743i)T
3 1+(−0.309+0.951i)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+(0.994−0.104i)T2 1 + (0.994 - 0.104i)T^{2}
11 1−iT2 1 - iT^{2}
13 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1+(−0.692+1.80i)T+(−0.743−0.669i)T2 1 + (-0.692 + 1.80i)T + (-0.743 - 0.669i)T^{2}
19 1+(−0.278−0.309i)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+(0.951−0.309i)T2 1 + (1.65 - 0.262i)T + (0.951 - 0.309i)T^{2}
29 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
31 1+(−0.685−1.05i)T+(−0.406+0.913i)T2 1 + (-0.685 - 1.05i)T + (-0.406 + 0.913i)T^{2}
37 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
41 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
43 1+(0.743−0.669i)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+(−0.406−0.913i)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+(0.978−0.207i)T2 1 + (0.978 - 0.207i)T^{2}
79 1+(1.86−0.715i)T+(0.743−0.669i)T2 1 + (1.86 - 0.715i)T + (0.743 - 0.669i)T^{2}
83 1+(0.413+1.94i)T+(−0.913+0.406i)T2 1 + (0.413 + 1.94i)T + (-0.913 + 0.406i)T^{2}
89 1+(0.587−0.809i)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(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

−8.520835418675228056055675298903, −7.48576775251704706060254663148, −6.82841026589873577202575760372, −5.97362261223308726267977152087, −5.53704208226566623344243482955, −4.62780473204381418907939399088, −3.37603191978107673667251945673, −2.73514877329721288835417982230, −1.93930216876560395610471382687, −1.00939866272458158434239695742, 1.99732315808733553506832139194, 2.87921827755711657415228620231, 3.91773278623148982903527795648, 4.36297608999466799395601712396, 5.50881907575141353155388127412, 5.78538132333117834917616725708, 6.51795405663168902208065540479, 7.64163484833410241200954920387, 8.370292709270745375902396691453, 8.788994492071571794328288351098

Graph of the ZZ-function along the critical line