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Properties

Label 2-1110-1.1-c3-0-70
Degree 22
Conductor 11101110
Sign −1-1
Analytic cond. 65.492165.4921
Root an. cond. 8.092728.09272
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 4·4-s + 5·5-s + 6·6-s − 15·7-s + 8·8-s + 9·9-s + 10·10-s − 50.2·11-s + 12·12-s + 21.0·13-s − 30·14-s + 15·15-s + 16·16-s − 52.0·17-s + 18·18-s − 101.·19-s + 20·20-s − 45·21-s − 100.·22-s + 37.5·23-s + 24·24-s + 25·25-s + 42.1·26-s + 27·27-s − 60·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.809·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.37·11-s + 0.288·12-s + 0.449·13-s − 0.572·14-s + 0.258·15-s + 0.250·16-s − 0.742·17-s + 0.235·18-s − 1.22·19-s + 0.223·20-s − 0.467·21-s − 0.973·22-s + 0.340·23-s + 0.204·24-s + 0.200·25-s + 0.317·26-s + 0.192·27-s − 0.404·28-s + ⋯

Functional equation

Λ(s)=(1110s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1110s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11101110    =    2⋅3⋅5⋅372 \cdot 3 \cdot 5 \cdot 37
Sign: −1-1
Analytic conductor: 65.492165.4921
Root analytic conductor: 8.092728.09272
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1110, ( :3/2), −1)(2,\ 1110,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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 1−2T 1 - 2T
3 1−3T 1 - 3T
5 1−5T 1 - 5T
37 1+37T 1 + 37T
good7 1+15T+343T2 1 + 15T + 343T^{2}
11 1+50.2T+1.33e3T2 1 + 50.2T + 1.33e3T^{2}
13 1−21.0T+2.19e3T2 1 - 21.0T + 2.19e3T^{2}
17 1+52.0T+4.91e3T2 1 + 52.0T + 4.91e3T^{2}
19 1+101.T+6.85e3T2 1 + 101.T + 6.85e3T^{2}
23 1−37.5T+1.21e4T2 1 - 37.5T + 1.21e4T^{2}
29 1+217.T+2.43e4T2 1 + 217.T + 2.43e4T^{2}
31 1+92.4T+2.97e4T2 1 + 92.4T + 2.97e4T^{2}
41 1−418.T+6.89e4T2 1 - 418.T + 6.89e4T^{2}
43 1+383.T+7.95e4T2 1 + 383.T + 7.95e4T^{2}
47 1+10.5T+1.03e5T2 1 + 10.5T + 1.03e5T^{2}
53 1+692.T+1.48e5T2 1 + 692.T + 1.48e5T^{2}
59 1−604.T+2.05e5T2 1 - 604.T + 2.05e5T^{2}
61 1−256.T+2.26e5T2 1 - 256.T + 2.26e5T^{2}
67 1+211.T+3.00e5T2 1 + 211.T + 3.00e5T^{2}
71 1+385.T+3.57e5T2 1 + 385.T + 3.57e5T^{2}
73 1+102.T+3.89e5T2 1 + 102.T + 3.89e5T^{2}
79 1+131.T+4.93e5T2 1 + 131.T + 4.93e5T^{2}
83 1+190.T+5.71e5T2 1 + 190.T + 5.71e5T^{2}
89 1+1.08e3T+7.04e5T2 1 + 1.08e3T + 7.04e5T^{2}
97 1−1.45e3T+9.12e5T2 1 - 1.45e3T + 9.12e5T^{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

−9.067390319827868482280830045114, −8.215241971356809728310252408823, −7.27534755602640002920595316717, −6.43016238123071192543092571516, −5.64883855075278718172805936018, −4.65056697063125303034925956830, −3.62312777389315297434712970495, −2.71833802875145940837054996106, −1.88738428039713629892708528973, 0, 1.88738428039713629892708528973, 2.71833802875145940837054996106, 3.62312777389315297434712970495, 4.65056697063125303034925956830, 5.64883855075278718172805936018, 6.43016238123071192543092571516, 7.27534755602640002920595316717, 8.215241971356809728310252408823, 9.067390319827868482280830045114

Graph of the ZZ-function along the critical line