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Properties

Label 2-3e5-1.1-c1-0-6
Degree 22
Conductor 243243
Sign 11
Analytic cond. 1.940361.94036
Root an. cond. 1.392961.39296
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  + 1.73·2-s + 0.999·4-s + 3.46·5-s − 7-s − 1.73·8-s + 5.99·10-s − 3.46·11-s + 5·13-s − 1.73·14-s − 5·16-s − 19-s + 3.46·20-s − 5.99·22-s − 6.92·23-s + 6.99·25-s + 8.66·26-s − 0.999·28-s − 3.46·29-s + 5·31-s − 5.19·32-s − 3.46·35-s − 37-s − 1.73·38-s − 6.00·40-s + 3.46·41-s − 43-s − 3.46·44-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.499·4-s + 1.54·5-s − 0.377·7-s − 0.612·8-s + 1.89·10-s − 1.04·11-s + 1.38·13-s − 0.462·14-s − 1.25·16-s − 0.229·19-s + 0.774·20-s − 1.27·22-s − 1.44·23-s + 1.39·25-s + 1.69·26-s − 0.188·28-s − 0.643·29-s + 0.898·31-s − 0.918·32-s − 0.585·35-s − 0.164·37-s − 0.280·38-s − 0.948·40-s + 0.541·41-s − 0.152·43-s − 0.522·44-s + ⋯

Functional equation

Λ(s)=(243s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(243s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 243243    =    353^{5}
Sign: 11
Analytic conductor: 1.940361.94036
Root analytic conductor: 1.392961.39296
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 243, ( :1/2), 1)(2,\ 243,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3762162782.376216278
L(12)L(\frac12) ≈\approx 2.3762162782.376216278
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 1
good2 1−1.73T+2T2 1 - 1.73T + 2T^{2}
5 1−3.46T+5T2 1 - 3.46T + 5T^{2}
7 1+T+7T2 1 + T + 7T^{2}
11 1+3.46T+11T2 1 + 3.46T + 11T^{2}
13 1−5T+13T2 1 - 5T + 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1+T+19T2 1 + T + 19T^{2}
23 1+6.92T+23T2 1 + 6.92T + 23T^{2}
29 1+3.46T+29T2 1 + 3.46T + 29T^{2}
31 1−5T+31T2 1 - 5T + 31T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1−3.46T+41T2 1 - 3.46T + 41T^{2}
43 1+T+43T2 1 + T + 43T^{2}
47 1+3.46T+47T2 1 + 3.46T + 47T^{2}
53 1+10.3T+53T2 1 + 10.3T + 53T^{2}
59 1−3.46T+59T2 1 - 3.46T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1−8T+67T2 1 - 8T + 67T^{2}
71 1−10.3T+71T2 1 - 10.3T + 71T^{2}
73 1−2T+73T2 1 - 2T + 73T^{2}
79 1+T+79T2 1 + T + 79T^{2}
83 1−6.92T+83T2 1 - 6.92T + 83T^{2}
89 1−10.3T+89T2 1 - 10.3T + 89T^{2}
97 1−17T+97T2 1 - 17T + 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

−12.60055799547426009652810274468, −11.27363637433942366154952747032, −10.21723916378989491163277649923, −9.388168687038182801682601425010, −8.223590322904053967862384369796, −6.38988047348155697162740997975, −5.94367436151325889775645239570, −4.95763268889965373286462212722, −3.52168957571220695631214088920, −2.21095194874431803639976263612, 2.21095194874431803639976263612, 3.52168957571220695631214088920, 4.95763268889965373286462212722, 5.94367436151325889775645239570, 6.38988047348155697162740997975, 8.223590322904053967862384369796, 9.388168687038182801682601425010, 10.21723916378989491163277649923, 11.27363637433942366154952747032, 12.60055799547426009652810274468

Graph of the ZZ-function along the critical line