↑

Properties

Label 2-11-1.1-c1-0-0
Degree 22
Conductor 1111
Sign 11
Analytic cond. 0.08783540.0878354
Root an. cond. 0.2963700.296370
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s − 2·7-s − 2·9-s − 2·10-s + 11-s − 2·12-s + 4·13-s + 4·14-s − 15-s − 4·16-s − 2·17-s + 4·18-s + 2·20-s + 2·21-s − 2·22-s − 23-s − 4·25-s − 8·26-s + 5·27-s − 4·28-s + 2·30-s + 7·31-s + 8·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s − 2/3·9-s − 0.632·10-s + 0.301·11-s − 0.577·12-s + 1.10·13-s + 1.06·14-s − 0.258·15-s − 16-s − 0.485·17-s + 0.942·18-s + 0.447·20-s + 0.436·21-s − 0.426·22-s − 0.208·23-s − 4/5·25-s − 1.56·26-s + 0.962·27-s − 0.755·28-s + 0.365·30-s + 1.25·31-s + 1.41·32-s + ⋯

Functional equation

Λ(s)=(11s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(11s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1111
Sign: 11
Analytic conductor: 0.08783540.0878354
Root analytic conductor: 0.2963700.296370
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 11, ( :1/2), 1)(2,\ 11,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.25384186080.2538418608
L(12)L(\frac12) ≈\approx 0.25384186080.2538418608
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
bad11 1−T 1 - T
good2 1+pT+pT2 1 + p T + p T^{2} 1.2.c
3 1+T+pT2 1 + T + p T^{2} 1.3.b
5 1−T+pT2 1 - T + p T^{2} 1.5.ab
7 1+2T+pT2 1 + 2 T + p T^{2} 1.7.c
13 1−4T+pT2 1 - 4 T + p T^{2} 1.13.ae
17 1+2T+pT2 1 + 2 T + p T^{2} 1.17.c
19 1+pT2 1 + p T^{2} 1.19.a
23 1+T+pT2 1 + T + p T^{2} 1.23.b
29 1+pT2 1 + p T^{2} 1.29.a
31 1−7T+pT2 1 - 7 T + p T^{2} 1.31.ah
37 1−3T+pT2 1 - 3 T + p T^{2} 1.37.ad
41 1+8T+pT2 1 + 8 T + p T^{2} 1.41.i
43 1+6T+pT2 1 + 6 T + p T^{2} 1.43.g
47 1−8T+pT2 1 - 8 T + p T^{2} 1.47.ai
53 1+6T+pT2 1 + 6 T + p T^{2} 1.53.g
59 1−5T+pT2 1 - 5 T + p T^{2} 1.59.af
61 1−12T+pT2 1 - 12 T + p T^{2} 1.61.am
67 1+7T+pT2 1 + 7 T + p T^{2} 1.67.h
71 1+3T+pT2 1 + 3 T + p T^{2} 1.71.d
73 1−4T+pT2 1 - 4 T + p T^{2} 1.73.ae
79 1+10T+pT2 1 + 10 T + p T^{2} 1.79.k
83 1+6T+pT2 1 + 6 T + p T^{2} 1.83.g
89 1−15T+pT2 1 - 15 T + p T^{2} 1.89.ap
97 1+7T+pT2 1 + 7 T + p T^{2} 1.97.h
show more
show less
   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

−20.37926046435010943649954977728, −19.18572497185224141236190272085, −17.94143357345934104539279635252, −17.03361032038062445390026939405, −15.91407260330038416331639924088, −13.56863905712999451342489435843, −11.45125861034521058353374083886, −10.03550909718107888433464868208, −8.603539619290756001226038948684, −6.36261389471308870138602900888, 6.36261389471308870138602900888, 8.603539619290756001226038948684, 10.03550909718107888433464868208, 11.45125861034521058353374083886, 13.56863905712999451342489435843, 15.91407260330038416331639924088, 17.03361032038062445390026939405, 17.94143357345934104539279635252, 19.18572497185224141236190272085, 20.37926046435010943649954977728

Graph of the ZZ-function along the critical line