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Properties

Label 2-192-3.2-c2-0-8
Degree 22
Conductor 192192
Sign 11
Analytic cond. 5.231625.23162
Root an. cond. 2.287272.28727
Motivic weight 22
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 2·7-s + 9·9-s + 22·13-s − 26·19-s + 6·21-s + 25·25-s + 27·27-s − 46·31-s − 26·37-s + 66·39-s + 22·43-s − 45·49-s − 78·57-s − 74·61-s + 18·63-s − 122·67-s − 46·73-s + 75·75-s − 142·79-s + 81·81-s + 44·91-s − 138·93-s + 2·97-s + 194·103-s + 214·109-s − 78·111-s + ⋯
L(s)  = 1  + 3-s + 2/7·7-s + 9-s + 1.69·13-s − 1.36·19-s + 2/7·21-s + 25-s + 27-s − 1.48·31-s − 0.702·37-s + 1.69·39-s + 0.511·43-s − 0.918·49-s − 1.36·57-s − 1.21·61-s + 2/7·63-s − 1.82·67-s − 0.630·73-s + 75-s − 1.79·79-s + 81-s + 0.483·91-s − 1.48·93-s + 2/97·97-s + 1.88·103-s + 1.96·109-s − 0.702·111-s + ⋯

Functional equation

Λ(s)=(192s/2 ΓC(s) L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(192s/2 ΓC(s+1) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 192192    =    26⋅32^{6} \cdot 3
Sign: 11
Analytic conductor: 5.231625.23162
Root analytic conductor: 2.287272.28727
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: χ192(65,⋅)\chi_{192} (65, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 192, ( :1), 1)(2,\ 192,\ (\ :1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.2083262952.208326295
L(12)L(\frac12) ≈\approx 2.2083262952.208326295
L(2)L(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 1
3 1−pT 1 - p T
good5 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
7 1−2T+p2T2 1 - 2 T + p^{2} T^{2}
11 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
13 1−22T+p2T2 1 - 22 T + p^{2} T^{2}
17 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
19 1+26T+p2T2 1 + 26 T + p^{2} T^{2}
23 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
29 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
31 1+46T+p2T2 1 + 46 T + p^{2} T^{2}
37 1+26T+p2T2 1 + 26 T + p^{2} T^{2}
41 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
43 1−22T+p2T2 1 - 22 T + p^{2} T^{2}
47 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
53 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
59 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
61 1+74T+p2T2 1 + 74 T + p^{2} T^{2}
67 1+122T+p2T2 1 + 122 T + p^{2} T^{2}
71 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
73 1+46T+p2T2 1 + 46 T + p^{2} T^{2}
79 1+142T+p2T2 1 + 142 T + p^{2} T^{2}
83 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
89 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
97 1−2T+p2T2 1 - 2 T + p^{2} T^{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

−12.61473187465100986317566421057, −11.13112602429135677172967195379, −10.39382314282203396087111061641, −8.959671153175178664032384476176, −8.524206686848126024137066396593, −7.31461290749082860154761378751, −6.11540748883271152272567997396, −4.44365225141518026083351121884, −3.30800484609230149778015279707, −1.67565795327991440860324882684, 1.67565795327991440860324882684, 3.30800484609230149778015279707, 4.44365225141518026083351121884, 6.11540748883271152272567997396, 7.31461290749082860154761378751, 8.524206686848126024137066396593, 8.959671153175178664032384476176, 10.39382314282203396087111061641, 11.13112602429135677172967195379, 12.61473187465100986317566421057

Graph of the ZZ-function along the critical line