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Properties

Label 2-273-1.1-c1-0-9
Degree 22
Conductor 273273
Sign 11
Analytic cond. 2.179912.17991
Root an. cond. 1.476451.47645
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  + 2.61·2-s + 3-s + 4.81·4-s − 3.81·5-s + 2.61·6-s + 7-s + 7.34·8-s + 9-s − 9.95·10-s − 4.73·11-s + 4.81·12-s + 13-s + 2.61·14-s − 3.81·15-s + 9.55·16-s − 5.22·17-s + 2.61·18-s + 2.92·19-s − 18.3·20-s + 21-s − 12.3·22-s + 3.33·23-s + 7.34·24-s + 9.55·25-s + 2.61·26-s + 27-s + 4.81·28-s + ⋯
L(s)  = 1  + 1.84·2-s + 0.577·3-s + 2.40·4-s − 1.70·5-s + 1.06·6-s + 0.377·7-s + 2.59·8-s + 0.333·9-s − 3.14·10-s − 1.42·11-s + 1.38·12-s + 0.277·13-s + 0.697·14-s − 0.984·15-s + 2.38·16-s − 1.26·17-s + 0.615·18-s + 0.670·19-s − 4.10·20-s + 0.218·21-s − 2.63·22-s + 0.694·23-s + 1.49·24-s + 1.91·25-s + 0.511·26-s + 0.192·27-s + 0.909·28-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), 1)(2,\ 273,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2556605123.255660512
L(12)L(\frac12) ≈\approx 3.2556605123.255660512
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
13 1−T 1 - T
good2 1−2.61T+2T2 1 - 2.61T + 2T^{2}
5 1+3.81T+5T2 1 + 3.81T + 5T^{2}
11 1+4.73T+11T2 1 + 4.73T + 11T^{2}
17 1+5.22T+17T2 1 + 5.22T + 17T^{2}
19 1−2.92T+19T2 1 - 2.92T + 19T^{2}
23 1−3.33T+23T2 1 - 3.33T + 23T^{2}
29 1+0.922T+29T2 1 + 0.922T + 29T^{2}
31 1+7.51T+31T2 1 + 7.51T + 31T^{2}
37 1−0.154T+37T2 1 - 0.154T + 37T^{2}
41 1−6.36T+41T2 1 - 6.36T + 41T^{2}
43 1+6.55T+43T2 1 + 6.55T + 43T^{2}
47 1−9.03T+47T2 1 - 9.03T + 47T^{2}
53 1−8.55T+53T2 1 - 8.55T + 53T^{2}
59 1−3.95T+59T2 1 - 3.95T + 59T^{2}
61 1−12.4T+61T2 1 - 12.4T + 61T^{2}
67 1+10.6T+67T2 1 + 10.6T + 67T^{2}
71 1+6.58T+71T2 1 + 6.58T + 71T^{2}
73 1+7.73T+73T2 1 + 7.73T + 73T^{2}
79 1−13.3T+79T2 1 - 13.3T + 79T^{2}
83 1+1.40T+83T2 1 + 1.40T + 83T^{2}
89 1+1.96T+89T2 1 + 1.96T + 89T^{2}
97 1+2.11T+97T2 1 + 2.11T + 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

−12.09501066929474667624338485190, −11.23939024306669450237144544528, −10.70243886733941868471052610459, −8.665553729300073867917416106964, −7.59812598639803922425882860881, −7.07590021453430569261566645838, −5.41369456039447429979351854216, −4.47083612524782670720743209297, −3.62387459756726331630318828744, −2.55677068828348872473276058447, 2.55677068828348872473276058447, 3.62387459756726331630318828744, 4.47083612524782670720743209297, 5.41369456039447429979351854216, 7.07590021453430569261566645838, 7.59812598639803922425882860881, 8.665553729300073867917416106964, 10.70243886733941868471052610459, 11.23939024306669450237144544528, 12.09501066929474667624338485190

Graph of the ZZ-function along the critical line