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Properties

Label 2-186-1.1-c1-0-4
Degree 22
Conductor 186186
Sign 11
Analytic cond. 1.485211.48521
Root an. cond. 1.218691.21869
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 − 2·7-s + 8-s + 9-s + 10-s − 3·11-s + 12-s − 13-s − 2·14-s + 15-s + 16-s + 3·17-s + 18-s − 5·19-s + 20-s − 2·21-s − 3·22-s + 4·23-s + 24-s − 4·25-s − 26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.436·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.377·28-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: yes
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 1.9528976681.952897668
L(12)L(\frac12) ≈\approx 1.9528976681.952897668
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
bad2 1−T 1 - T
3 1−T 1 - T
31 1−T 1 - T
good5 1−T+pT2 1 - T + p T^{2} 1.5.ab
7 1+2T+pT2 1 + 2 T + p T^{2} 1.7.c
11 1+3T+pT2 1 + 3 T + p T^{2} 1.11.d
13 1+T+pT2 1 + T + p T^{2} 1.13.b
17 1−3T+pT2 1 - 3 T + p T^{2} 1.17.ad
19 1+5T+pT2 1 + 5 T + p T^{2} 1.19.f
23 1−4T+pT2 1 - 4 T + p T^{2} 1.23.ae
29 1+pT2 1 + p T^{2} 1.29.a
37 1+2T+pT2 1 + 2 T + p T^{2} 1.37.c
41 1−2T+pT2 1 - 2 T + p T^{2} 1.41.ac
43 1+6T+pT2 1 + 6 T + p T^{2} 1.43.g
47 1+7T+pT2 1 + 7 T + p T^{2} 1.47.h
53 1−14T+pT2 1 - 14 T + p T^{2} 1.53.ao
59 1−10T+pT2 1 - 10 T + p T^{2} 1.59.ak
61 1−7T+pT2 1 - 7 T + p T^{2} 1.61.ah
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+6T+pT2 1 + 6 T + p T^{2} 1.73.g
79 1−15T+pT2 1 - 15 T + p T^{2} 1.79.ap
83 1+T+pT2 1 + T + p T^{2} 1.83.b
89 1−10T+pT2 1 - 10 T + p T^{2} 1.89.ak
97 1−13T+pT2 1 - 13 T + p T^{2} 1.97.an
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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

−13.05040591723050562208382603170, −11.86609793920404254157071370869, −10.48645678616775104865148338230, −9.801450505814553909137448242559, −8.512584183321875206998435933939, −7.33319367325295365011876797445, −6.21449897841157477524696518952, −5.03620516471027669405867234682, −3.54463186962642716865631176024, −2.34546937125578111204655036909, 2.34546937125578111204655036909, 3.54463186962642716865631176024, 5.03620516471027669405867234682, 6.21449897841157477524696518952, 7.33319367325295365011876797445, 8.512584183321875206998435933939, 9.801450505814553909137448242559, 10.48645678616775104865148338230, 11.86609793920404254157071370869, 13.05040591723050562208382603170

Graph of the ZZ-function along the critical line