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Properties

Label 2-483-1.1-c1-0-20
Degree 22
Conductor 483483
Sign 11
Analytic cond. 3.856773.85677
Root an. cond. 1.963861.96386
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s + 4·5-s + 2·6-s − 7-s + 9-s + 8·10-s − 5·11-s + 2·12-s − 2·13-s − 2·14-s + 4·15-s − 4·16-s + 2·18-s − 5·19-s + 8·20-s − 21-s − 10·22-s − 23-s + 11·25-s − 4·26-s + 27-s − 2·28-s − 2·29-s + 8·30-s + 6·31-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 1.78·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 2.52·10-s − 1.50·11-s + 0.577·12-s − 0.554·13-s − 0.534·14-s + 1.03·15-s − 16-s + 0.471·18-s − 1.14·19-s + 1.78·20-s − 0.218·21-s − 2.13·22-s − 0.208·23-s + 11/5·25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + 1.46·30-s + 1.07·31-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 11
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 1)(2,\ 483,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.7674424073.767442407
L(12)L(\frac12) ≈\approx 3.7674424073.767442407
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)
bad3 1−T 1 - T
7 1+T 1 + T
23 1+T 1 + T
good2 1−pT+pT2 1 - p T + p T^{2}
5 1−4T+pT2 1 - 4 T + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1−9T+pT2 1 - 9 T + p T^{2}
61 1+5T+pT2 1 + 5 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+pT2 1 + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1+18T+pT2 1 + 18 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+18T+pT2 1 + 18 T + p 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

−10.94122298294594700260446625869, −10.06348584847165124277699778695, −9.421863300952081605429638012938, −8.292339792365995708433110594415, −6.91642082849532995462484101470, −6.00450627059691435421291295880, −5.33569718385275672521817298149, −4.33710151569730855874203864959, −2.74710044114612301763782652477, −2.32541152601280590701225190627, 2.32541152601280590701225190627, 2.74710044114612301763782652477, 4.33710151569730855874203864959, 5.33569718385275672521817298149, 6.00450627059691435421291295880, 6.91642082849532995462484101470, 8.292339792365995708433110594415, 9.421863300952081605429638012938, 10.06348584847165124277699778695, 10.94122298294594700260446625869

Graph of the ZZ-function along the critical line