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Properties

Label 2-141-1.1-c1-0-2
Degree 22
Conductor 141141
Sign 11
Analytic cond. 1.125891.12589
Root an. cond. 1.061071.06107
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-s + 3-s − 4-s + 2·5-s − 6-s + 3·8-s + 9-s − 2·10-s + 4·11-s − 12-s − 2·13-s + 2·15-s − 16-s + 2·17-s − 18-s − 2·20-s − 4·22-s + 3·24-s − 25-s + 2·26-s + 27-s − 6·29-s − 2·30-s − 4·31-s − 5·32-s + 4·33-s − 2·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.447·20-s − 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 1.11·29-s − 0.365·30-s − 0.718·31-s − 0.883·32-s + 0.696·33-s − 0.342·34-s + ⋯

Functional equation

Λ(s)=(141s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(141s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 141141    =    3⋅473 \cdot 47
Sign: 11
Analytic conductor: 1.125891.12589
Root analytic conductor: 1.061071.06107
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 141, ( :1/2), 1)(2,\ 141,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.96247796910.9624779691
L(12)L(\frac12) ≈\approx 0.96247796910.9624779691
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)Isogeny Class over Fp\mathbf{F}_p
bad3 1−T 1 - T
47 1+T 1 + T
good2 1+T+pT2 1 + T + p T^{2} 1.2.b
5 1−2T+pT2 1 - 2 T + p T^{2} 1.5.ac
7 1+pT2 1 + p T^{2} 1.7.a
11 1−4T+pT2 1 - 4 T + p T^{2} 1.11.ae
13 1+2T+pT2 1 + 2 T + p T^{2} 1.13.c
17 1−2T+pT2 1 - 2 T + p T^{2} 1.17.ac
19 1+pT2 1 + p T^{2} 1.19.a
23 1+pT2 1 + p T^{2} 1.23.a
29 1+6T+pT2 1 + 6 T + p T^{2} 1.29.g
31 1+4T+pT2 1 + 4 T + p T^{2} 1.31.e
37 1+10T+pT2 1 + 10 T + p T^{2} 1.37.k
41 1+2T+pT2 1 + 2 T + p T^{2} 1.41.c
43 1−8T+pT2 1 - 8 T + p T^{2} 1.43.ai
53 1+2T+pT2 1 + 2 T + p T^{2} 1.53.c
59 1+4T+pT2 1 + 4 T + p T^{2} 1.59.e
61 1−14T+pT2 1 - 14 T + p T^{2} 1.61.ao
67 1+8T+pT2 1 + 8 T + p T^{2} 1.67.i
71 1−16T+pT2 1 - 16 T + p T^{2} 1.71.aq
73 1−2T+pT2 1 - 2 T + p T^{2} 1.73.ac
79 1−8T+pT2 1 - 8 T + p T^{2} 1.79.ai
83 1+4T+pT2 1 + 4 T + p T^{2} 1.83.e
89 1−18T+pT2 1 - 18 T + p T^{2} 1.89.as
97 1+14T+pT2 1 + 14 T + p T^{2} 1.97.o
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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

−13.33645536469427920089901573537, −12.25269126968811006373592111535, −10.75662405333874085581848740520, −9.575443434008070535852551553455, −9.310688144069273468867229042449, −8.104359289936234206684837124367, −6.91995908324451733419004550129, −5.36270443497112201719182264503, −3.82986653803997716904500612796, −1.74630150225815909613391524573, 1.74630150225815909613391524573, 3.82986653803997716904500612796, 5.36270443497112201719182264503, 6.91995908324451733419004550129, 8.104359289936234206684837124367, 9.310688144069273468867229042449, 9.575443434008070535852551553455, 10.75662405333874085581848740520, 12.25269126968811006373592111535, 13.33645536469427920089901573537

Graph of the ZZ-function along the critical line