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Properties

Label 2-192-8.5-c1-0-3
Degree 22
Conductor 192192
Sign −0.258+0.965i-0.258 + 0.965i
Analytic cond. 1.533121.53312
Root an. cond. 1.238191.23819
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s − 3.46i·5-s − 3.46·7-s − 9-s − 3.46·15-s + 6·17-s − 4i·19-s + 3.46i·21-s + 6.92·23-s − 6.99·25-s + i·27-s − 3.46i·29-s + 3.46·31-s + 11.9i·35-s + 6.92i·37-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.54i·5-s − 1.30·7-s − 0.333·9-s − 0.894·15-s + 1.45·17-s − 0.917i·19-s + 0.755i·21-s + 1.44·23-s − 1.39·25-s + 0.192i·27-s − 0.643i·29-s + 0.622·31-s + 2.02i·35-s + 1.13i·37-s + ⋯

Functional equation

Λ(s)=(192s/2 ΓC(s) L(s)=((−0.258+0.965i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(192s/2 ΓC(s+1/2) L(s)=((−0.258+0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 192192    =    26⋅32^{6} \cdot 3
Sign: −0.258+0.965i-0.258 + 0.965i
Analytic conductor: 1.533121.53312
Root analytic conductor: 1.238191.23819
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ192(97,⋅)\chi_{192} (97, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 192, ( :1/2), −0.258+0.965i)(2,\ 192,\ (\ :1/2),\ -0.258 + 0.965i)

Particular Values

L(1)L(1) ≈\approx 0.606131−0.789925i0.606131 - 0.789925i
L(12)L(\frac12) ≈\approx 0.606131−0.789925i0.606131 - 0.789925i
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 1
3 1+iT 1 + iT
good5 1+3.46iT−5T2 1 + 3.46iT - 5T^{2}
7 1+3.46T+7T2 1 + 3.46T + 7T^{2}
11 1−11T2 1 - 11T^{2}
13 1−13T2 1 - 13T^{2}
17 1−6T+17T2 1 - 6T + 17T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1−6.92T+23T2 1 - 6.92T + 23T^{2}
29 1+3.46iT−29T2 1 + 3.46iT - 29T^{2}
31 1−3.46T+31T2 1 - 3.46T + 31T^{2}
37 1−6.92iT−37T2 1 - 6.92iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1−6.92T+47T2 1 - 6.92T + 47T^{2}
53 1−3.46iT−53T2 1 - 3.46iT - 53T^{2}
59 1+12iT−59T2 1 + 12iT - 59T^{2}
61 1−6.92iT−61T2 1 - 6.92iT - 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1+6.92T+71T2 1 + 6.92T + 71T^{2}
73 1−2T+73T2 1 - 2T + 73T^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1+2T+97T2 1 + 2T + 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

−12.48267599121218769925128680887, −11.62827665089967695760498063079, −10.02485071710212256049631450457, −9.201938143900970063825023534401, −8.314869136830471831185090861616, −7.08786237725335792325890154552, −5.90217453832004117775645133580, −4.78591559504293675955562833554, −3.10614619104068965326720920865, −0.941758406426872705971917835081, 2.94859128859384174501581603221, 3.62548604982906873751766525295, 5.59151821386866085061786196819, 6.61017659790386139802871915396, 7.52074758091274109258973991389, 9.095635300961643552420937018514, 10.16891608252132565382441305515, 10.50710315024056900055444903250, 11.74748520578607004936442968860, 12.78365828156245111806504779527

Graph of the ZZ-function along the critical line