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Properties

Label 2-483-1.1-c1-0-18
Degree 22
Conductor 483483
Sign −1-1
Analytic cond. 3.856773.85677
Root an. cond. 1.963861.96386
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.381·2-s + 3-s − 1.85·4-s − 1.38·5-s − 0.381·6-s + 7-s + 1.47·8-s + 9-s + 0.527·10-s − 5.47·11-s − 1.85·12-s − 2.38·13-s − 0.381·14-s − 1.38·15-s + 3.14·16-s − 17-s − 0.381·18-s − 3·19-s + 2.56·20-s + 21-s + 2.09·22-s − 23-s + 1.47·24-s − 3.09·25-s + 0.909·26-s + 27-s − 1.85·28-s + ⋯
L(s)  = 1  − 0.270·2-s + 0.577·3-s − 0.927·4-s − 0.618·5-s − 0.155·6-s + 0.377·7-s + 0.520·8-s + 0.333·9-s + 0.166·10-s − 1.64·11-s − 0.535·12-s − 0.660·13-s − 0.102·14-s − 0.356·15-s + 0.786·16-s − 0.242·17-s − 0.0900·18-s − 0.688·19-s + 0.572·20-s + 0.218·21-s + 0.445·22-s − 0.208·23-s + 0.300·24-s − 0.618·25-s + 0.178·26-s + 0.192·27-s − 0.350·28-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: −1-1
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 483, ( :1/2), −1)(2,\ 483,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+0.381T+2T2 1 + 0.381T + 2T^{2}
5 1+1.38T+5T2 1 + 1.38T + 5T^{2}
11 1+5.47T+11T2 1 + 5.47T + 11T^{2}
13 1+2.38T+13T2 1 + 2.38T + 13T^{2}
17 1+T+17T2 1 + T + 17T^{2}
19 1+3T+19T2 1 + 3T + 19T^{2}
29 1+7.47T+29T2 1 + 7.47T + 29T^{2}
31 1+3.76T+31T2 1 + 3.76T + 31T^{2}
37 1−1.47T+37T2 1 - 1.47T + 37T^{2}
41 1+4.70T+41T2 1 + 4.70T + 41T^{2}
43 1−8.09T+43T2 1 - 8.09T + 43T^{2}
47 1−1.70T+47T2 1 - 1.70T + 47T^{2}
53 1+3.38T+53T2 1 + 3.38T + 53T^{2}
59 1+6.14T+59T2 1 + 6.14T + 59T^{2}
61 1−13.7T+61T2 1 - 13.7T + 61T^{2}
67 1−4.14T+67T2 1 - 4.14T + 67T^{2}
71 1+3.90T+71T2 1 + 3.90T + 71T^{2}
73 1−2.70T+73T2 1 - 2.70T + 73T^{2}
79 1−0.527T+79T2 1 - 0.527T + 79T^{2}
83 1+3T+83T2 1 + 3T + 83T^{2}
89 1−3.14T+89T2 1 - 3.14T + 89T^{2}
97 1−5T+97T2 1 - 5T + 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

−10.39823081149817439046632654717, −9.578180479679418414146930744942, −8.639076508177464065000002641327, −7.87526213274663625327873522313, −7.40093200900812343193993508106, −5.57563154157219914651930065485, −4.66043629966342784974621871937, −3.69656774805218198534431793330, −2.23088730330174361858629621432, 0, 2.23088730330174361858629621432, 3.69656774805218198534431793330, 4.66043629966342784974621871937, 5.57563154157219914651930065485, 7.40093200900812343193993508106, 7.87526213274663625327873522313, 8.639076508177464065000002641327, 9.578180479679418414146930744942, 10.39823081149817439046632654717

Graph of the ZZ-function along the critical line