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Properties

Label 2-186-1.1-c1-0-3
Degree 22
Conductor 186186
Sign 11
Analytic cond. 1.485211.48521
Root an. cond. 1.218691.21869
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 − 3-s + 4-s + 3.56·5-s − 6-s − 3.12·7-s + 8-s + 9-s + 3.56·10-s + 1.56·11-s − 12-s + 3.56·13-s − 3.12·14-s − 3.56·15-s + 16-s − 6.68·17-s + 18-s + 1.56·19-s + 3.56·20-s + 3.12·21-s + 1.56·22-s − 8·23-s − 24-s + 7.68·25-s + 3.56·26-s − 27-s − 3.12·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.59·5-s − 0.408·6-s − 1.18·7-s + 0.353·8-s + 0.333·9-s + 1.12·10-s + 0.470·11-s − 0.288·12-s + 0.987·13-s − 0.834·14-s − 0.919·15-s + 0.250·16-s − 1.62·17-s + 0.235·18-s + 0.358·19-s + 0.796·20-s + 0.681·21-s + 0.332·22-s − 1.66·23-s − 0.204·24-s + 1.53·25-s + 0.698·26-s − 0.192·27-s − 0.590·28-s + ⋯

Functional equation

Λ(s)=(186s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(186s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 186186    =    2⋅3⋅312 \cdot 3 \cdot 31
Sign: 11
Analytic conductor: 1.485211.48521
Root analytic conductor: 1.218691.21869
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 186, ( :1/2), 1)(2,\ 186,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6753813201.675381320
L(12)L(\frac12) ≈\approx 1.6753813201.675381320
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
3 1+T 1 + T
31 1+T 1 + T
good5 1−3.56T+5T2 1 - 3.56T + 5T^{2}
7 1+3.12T+7T2 1 + 3.12T + 7T^{2}
11 1−1.56T+11T2 1 - 1.56T + 11T^{2}
13 1−3.56T+13T2 1 - 3.56T + 13T^{2}
17 1+6.68T+17T2 1 + 6.68T + 17T^{2}
19 1−1.56T+19T2 1 - 1.56T + 19T^{2}
23 1+8T+23T2 1 + 8T + 23T^{2}
29 1−1.12T+29T2 1 - 1.12T + 29T^{2}
37 1+8.24T+37T2 1 + 8.24T + 37T^{2}
41 1+12.2T+41T2 1 + 12.2T + 41T^{2}
43 1+0.876T+43T2 1 + 0.876T + 43T^{2}
47 1−8.68T+47T2 1 - 8.68T + 47T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1−7.12T+59T2 1 - 7.12T + 59T^{2}
61 1−3.56T+61T2 1 - 3.56T + 61T^{2}
67 1−12.6T+67T2 1 - 12.6T + 67T^{2}
71 1+5.56T+71T2 1 + 5.56T + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1−8.68T+79T2 1 - 8.68T + 79T^{2}
83 1+7.80T+83T2 1 + 7.80T + 83T^{2}
89 1−16.2T+89T2 1 - 16.2T + 89T^{2}
97 1−5.80T+97T2 1 - 5.80T + 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

−12.82970242883536565766697726296, −11.77635719893748808657083014272, −10.58074293327207340388459008405, −9.848792134156071187791953251953, −8.813422363596946949301763839220, −6.65478381999053354976868805730, −6.33537835797121285546506423012, −5.32146829320858463202368391376, −3.74481057082569996274433617289, −2.02882643892120175759435607569, 2.02882643892120175759435607569, 3.74481057082569996274433617289, 5.32146829320858463202368391376, 6.33537835797121285546506423012, 6.65478381999053354976868805730, 8.813422363596946949301763839220, 9.848792134156071187791953251953, 10.58074293327207340388459008405, 11.77635719893748808657083014272, 12.82970242883536565766697726296

Graph of the ZZ-function along the critical line