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Properties

Label 2-483-1.1-c1-0-2
Degree 22
Conductor 483483
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.69·2-s − 3-s + 0.868·4-s + 3.51·5-s + 1.69·6-s − 7-s + 1.91·8-s + 9-s − 5.94·10-s − 1.74·11-s − 0.868·12-s + 6.33·13-s + 1.69·14-s − 3.51·15-s − 4.98·16-s − 5.94·17-s − 1.69·18-s + 1.74·19-s + 3.04·20-s + 21-s + 2.95·22-s + 23-s − 1.91·24-s + 7.33·25-s − 10.7·26-s − 27-s − 0.868·28-s + ⋯
L(s)  = 1  − 1.19·2-s − 0.577·3-s + 0.434·4-s + 1.57·5-s + 0.691·6-s − 0.377·7-s + 0.677·8-s + 0.333·9-s − 1.88·10-s − 0.525·11-s − 0.250·12-s + 1.75·13-s + 0.452·14-s − 0.906·15-s − 1.24·16-s − 1.44·17-s − 0.399·18-s + 0.399·19-s + 0.681·20-s + 0.218·21-s + 0.629·22-s + 0.208·23-s − 0.391·24-s + 1.46·25-s − 2.10·26-s − 0.192·27-s − 0.164·28-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: 11
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: 00
Selberg data: (2, 483, ( :1/2), 1)(2,\ 483,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.79194225280.7919422528
L(12)L(\frac12) ≈\approx 0.79194225280.7919422528
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+1.69T+2T2 1 + 1.69T + 2T^{2}
5 1−3.51T+5T2 1 - 3.51T + 5T^{2}
11 1+1.74T+11T2 1 + 1.74T + 11T^{2}
13 1−6.33T+13T2 1 - 6.33T + 13T^{2}
17 1+5.94T+17T2 1 + 5.94T + 17T^{2}
19 1−1.74T+19T2 1 - 1.74T + 19T^{2}
29 1+5.68T+29T2 1 + 5.68T + 29T^{2}
31 1−7.94T+31T2 1 - 7.94T + 31T^{2}
37 1−1.53T+37T2 1 - 1.53T + 37T^{2}
41 1−12.1T+41T2 1 - 12.1T + 41T^{2}
43 1−6.43T+43T2 1 - 6.43T + 43T^{2}
47 1−3.59T+47T2 1 - 3.59T + 47T^{2}
53 1−12.9T+53T2 1 - 12.9T + 53T^{2}
59 1+8.69T+59T2 1 + 8.69T + 59T^{2}
61 1−8.47T+61T2 1 - 8.47T + 61T^{2}
67 1+4.46T+67T2 1 + 4.46T + 67T^{2}
71 1+13.4T+71T2 1 + 13.4T + 71T^{2}
73 1−4.29T+73T2 1 - 4.29T + 73T^{2}
79 1+7.42T+79T2 1 + 7.42T + 79T^{2}
83 1+7.75T+83T2 1 + 7.75T + 83T^{2}
89 1−4.18T+89T2 1 - 4.18T + 89T^{2}
97 1−5.53T+97T2 1 - 5.53T + 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

−10.71467008332142268420101746969, −10.06522826336291864322771359771, −9.183795088291003891715773775501, −8.697698004795462907973006810710, −7.37047365136649322702300170112, −6.30731291380189075104245450433, −5.70672545216908292895950509535, −4.33552640817428854407863321065, −2.36780115030343750314389151899, −1.07446734187228023635943245477, 1.07446734187228023635943245477, 2.36780115030343750314389151899, 4.33552640817428854407863321065, 5.70672545216908292895950509535, 6.30731291380189075104245450433, 7.37047365136649322702300170112, 8.697698004795462907973006810710, 9.183795088291003891715773775501, 10.06522826336291864322771359771, 10.71467008332142268420101746969

Graph of the ZZ-function along the critical line