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Properties

Label 2-110-1.1-c1-0-4
Degree 22
Conductor 110110
Sign 11
Analytic cond. 0.8783540.878354
Root an. cond. 0.9372050.937205
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 − 5-s + 6-s − 7-s + 8-s − 2·9-s − 10-s − 11-s + 12-s + 2·13-s − 14-s − 15-s + 16-s − 3·17-s − 2·18-s − 19-s − 20-s − 21-s − 22-s + 6·23-s + 24-s + 25-s + 2·26-s − 5·27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.229·19-s − 0.223·20-s − 0.218·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.962·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 0.8783540.878354
Root analytic conductor: 0.9372050.937205
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 110, ( :1/2), 1)(2,\ 110,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5407473441.540747344
L(12)L(\frac12) ≈\approx 1.5407473441.540747344
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)
bad2 1−T 1 - T
5 1+T 1 + T
11 1+T 1 + T
good3 1−T+pT2 1 - T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+9T+pT2 1 + 9 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1−5T+pT2 1 - 5 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−5T+pT2 1 - 5 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+9T+pT2 1 + 9 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−14T+pT2 1 - 14 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1−8T+pT2 1 - 8 T + p T^{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

−13.54712946528400016991259874830, −12.94269427172040951553295571447, −11.61430649233435892606346187577, −10.81523504438992298376035875160, −9.240245216843454718321617703983, −8.197325772660852886000765577503, −6.92339690043320565435471363407, −5.57585253794403166430858921429, −3.99189905466957435614609953053, −2.72390738347756879244011461463, 2.72390738347756879244011461463, 3.99189905466957435614609953053, 5.57585253794403166430858921429, 6.92339690043320565435471363407, 8.197325772660852886000765577503, 9.240245216843454718321617703983, 10.81523504438992298376035875160, 11.61430649233435892606346187577, 12.94269427172040951553295571447, 13.54712946528400016991259874830

Graph of the ZZ-function along the critical line