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Properties

Label 2-110-1.1-c1-0-2
Degree 22
Conductor 110110
Sign 11
Analytic cond. 0.8783540.878354
Root an. cond. 0.9372050.937205
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-s + 2.37·3-s + 4-s + 5-s − 2.37·6-s − 2.37·7-s − 8-s + 2.62·9-s − 10-s − 11-s + 2.37·12-s + 2·13-s + 2.37·14-s + 2.37·15-s + 16-s − 4.37·17-s − 2.62·18-s + 6.37·19-s + 20-s − 5.62·21-s + 22-s − 8.74·23-s − 2.37·24-s + 25-s − 2·26-s − 0.883·27-s − 2.37·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.36·3-s + 0.5·4-s + 0.447·5-s − 0.968·6-s − 0.896·7-s − 0.353·8-s + 0.875·9-s − 0.316·10-s − 0.301·11-s + 0.684·12-s + 0.554·13-s + 0.634·14-s + 0.612·15-s + 0.250·16-s − 1.06·17-s − 0.619·18-s + 1.46·19-s + 0.223·20-s − 1.22·21-s + 0.213·22-s − 1.82·23-s − 0.484·24-s + 0.200·25-s − 0.392·26-s − 0.169·27-s − 0.448·28-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: no
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 1.0929785021.092978502
L(12)L(\frac12) ≈\approx 1.0929785021.092978502
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 1−2.37T+3T2 1 - 2.37T + 3T^{2}
7 1+2.37T+7T2 1 + 2.37T + 7T^{2}
13 1−2T+13T2 1 - 2T + 13T^{2}
17 1+4.37T+17T2 1 + 4.37T + 17T^{2}
19 1−6.37T+19T2 1 - 6.37T + 19T^{2}
23 1+8.74T+23T2 1 + 8.74T + 23T^{2}
29 1+4.37T+29T2 1 + 4.37T + 29T^{2}
31 1+2.37T+31T2 1 + 2.37T + 31T^{2}
37 1−3.62T+37T2 1 - 3.62T + 37T^{2}
41 1−11.4T+41T2 1 - 11.4T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1+8.74T+47T2 1 + 8.74T + 47T^{2}
53 1−13.1T+53T2 1 - 13.1T + 53T^{2}
59 1−8.74T+59T2 1 - 8.74T + 59T^{2}
61 1−0.372T+61T2 1 - 0.372T + 61T^{2}
67 1−8T+67T2 1 - 8T + 67T^{2}
71 1+7.11T+71T2 1 + 7.11T + 71T^{2}
73 1−7.48T+73T2 1 - 7.48T + 73T^{2}
79 1+12.7T+79T2 1 + 12.7T + 79T^{2}
83 1−8.74T+83T2 1 - 8.74T + 83T^{2}
89 1−4.37T+89T2 1 - 4.37T + 89T^{2}
97 1+1.25T+97T2 1 + 1.25T + 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

−13.65791280221585508799586150832, −12.95801060937520669818156462953, −11.42377555288379445698982492618, −9.979494288438638459727049279184, −9.390247785900182046123628465206, −8.415837047933552989842722452006, −7.35422687001489073777346203906, −5.98353161137008289164431147583, −3.61519141027645554829414607861, −2.28619853611828694232914044186, 2.28619853611828694232914044186, 3.61519141027645554829414607861, 5.98353161137008289164431147583, 7.35422687001489073777346203906, 8.415837047933552989842722452006, 9.390247785900182046123628465206, 9.979494288438638459727049279184, 11.42377555288379445698982492618, 12.95801060937520669818156462953, 13.65791280221585508799586150832

Graph of the ZZ-function along the critical line