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Properties

Label 2-192-1.1-c1-0-1
Degree 22
Conductor 192192
Sign 11
Analytic cond. 1.533121.53312
Root an. cond. 1.238191.23819
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  + 3-s − 2·5-s + 4·7-s + 9-s + 4·11-s + 2·13-s − 2·15-s − 6·17-s − 4·19-s + 4·21-s − 25-s + 27-s − 2·29-s − 4·31-s + 4·33-s − 8·35-s + 2·37-s + 2·39-s + 2·41-s + 4·43-s − 2·45-s − 8·47-s + 9·49-s − 6·51-s − 10·53-s − 8·55-s − 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.696·33-s − 1.35·35-s + 0.328·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.840·51-s − 1.37·53-s − 1.07·55-s − 0.529·57-s + ⋯

Functional equation

Λ(s)=(192s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(192s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 192192    =    26⋅32^{6} \cdot 3
Sign: 11
Analytic conductor: 1.533121.53312
Root analytic conductor: 1.238191.23819
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 192, ( :1/2), 1)(2,\ 192,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4157372081.415737208
L(12)L(\frac12) ≈\approx 1.4157372081.415737208
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 1
3 1−T 1 - T
good5 1+2T+pT2 1 + 2 T + p T^{2} 1.5.c
7 1−4T+pT2 1 - 4 T + p T^{2} 1.7.ae
11 1−4T+pT2 1 - 4 T + p T^{2} 1.11.ae
13 1−2T+pT2 1 - 2 T + p T^{2} 1.13.ac
17 1+6T+pT2 1 + 6 T + p T^{2} 1.17.g
19 1+4T+pT2 1 + 4 T + p T^{2} 1.19.e
23 1+pT2 1 + p T^{2} 1.23.a
29 1+2T+pT2 1 + 2 T + p T^{2} 1.29.c
31 1+4T+pT2 1 + 4 T + p T^{2} 1.31.e
37 1−2T+pT2 1 - 2 T + p T^{2} 1.37.ac
41 1−2T+pT2 1 - 2 T + p T^{2} 1.41.ac
43 1−4T+pT2 1 - 4 T + p T^{2} 1.43.ae
47 1+8T+pT2 1 + 8 T + p T^{2} 1.47.i
53 1+10T+pT2 1 + 10 T + p T^{2} 1.53.k
59 1+4T+pT2 1 + 4 T + p T^{2} 1.59.e
61 1+6T+pT2 1 + 6 T + p T^{2} 1.61.g
67 1−4T+pT2 1 - 4 T + p T^{2} 1.67.ae
71 1−16T+pT2 1 - 16 T + p T^{2} 1.71.aq
73 1+6T+pT2 1 + 6 T + p T^{2} 1.73.g
79 1+4T+pT2 1 + 4 T + p T^{2} 1.79.e
83 1−12T+pT2 1 - 12 T + p T^{2} 1.83.am
89 1−10T+pT2 1 - 10 T + p T^{2} 1.89.ak
97 1+14T+pT2 1 + 14 T + p T^{2} 1.97.o
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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

−12.46429529504907365579234739579, −11.31959077603206060511694459208, −11.01151255843213055292894337270, −9.240123327709471504547415939693, −8.460175078145674768211752918036, −7.67780945046588747720337476950, −6.44896944953500655110243120870, −4.62570703678849832502580328887, −3.85566973953101374011358603281, −1.83689405023894510739841616557, 1.83689405023894510739841616557, 3.85566973953101374011358603281, 4.62570703678849832502580328887, 6.44896944953500655110243120870, 7.67780945046588747720337476950, 8.460175078145674768211752918036, 9.240123327709471504547415939693, 11.01151255843213055292894337270, 11.31959077603206060511694459208, 12.46429529504907365579234739579

Graph of the ZZ-function along the critical line