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Properties

Label 2-192-4.3-c8-0-15
Degree 22
Conductor 192192
Sign ii
Analytic cond. 78.216678.2166
Root an. cond. 8.844028.84402
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 46.7i·3-s − 726·5-s + 3.05e3i·7-s − 2.18e3·9-s + 1.32e4i·11-s − 3.90e4·13-s + 3.39e4i·15-s − 6.58e4·17-s − 1.30e5i·19-s + 1.42e5·21-s + 5.02e5i·23-s + 1.36e5·25-s + 1.02e5i·27-s − 2.02e5·29-s + 1.19e6i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.16·5-s + 1.27i·7-s − 0.333·9-s + 0.907i·11-s − 1.36·13-s + 0.670i·15-s − 0.787·17-s − 0.999i·19-s + 0.734·21-s + 1.79i·23-s + 0.349·25-s + 0.192i·27-s − 0.285·29-s + 1.29i·31-s + ⋯

Functional equation

Λ(s)=(192s/2 ΓC(s) L(s)=(i Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(192s/2 ΓC(s+4) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 192192    =    26⋅32^{6} \cdot 3
Sign: ii
Analytic conductor: 78.216678.2166
Root analytic conductor: 8.844028.84402
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ192(127,⋅)\chi_{192} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 192, ( :4), i)(2,\ 192,\ (\ :4),\ i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.37216633230.3721663323
L(12)L(\frac12) ≈\approx 0.37216633230.3721663323
L(5)L(5) 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+46.7iT 1 + 46.7iT
good5 1+726T+3.90e5T2 1 + 726T + 3.90e5T^{2}
7 1−3.05e3iT−5.76e6T2 1 - 3.05e3iT - 5.76e6T^{2}
11 1−1.32e4iT−2.14e8T2 1 - 1.32e4iT - 2.14e8T^{2}
13 1+3.90e4T+8.15e8T2 1 + 3.90e4T + 8.15e8T^{2}
17 1+6.58e4T+6.97e9T2 1 + 6.58e4T + 6.97e9T^{2}
19 1+1.30e5iT−1.69e10T2 1 + 1.30e5iT - 1.69e10T^{2}
23 1−5.02e5iT−7.83e10T2 1 - 5.02e5iT - 7.83e10T^{2}
29 1+2.02e5T+5.00e11T2 1 + 2.02e5T + 5.00e11T^{2}
31 1−1.19e6iT−8.52e11T2 1 - 1.19e6iT - 8.52e11T^{2}
37 1−1.87e6T+3.51e12T2 1 - 1.87e6T + 3.51e12T^{2}
41 1−3.09e6T+7.98e12T2 1 - 3.09e6T + 7.98e12T^{2}
43 1+2.26e6iT−1.16e13T2 1 + 2.26e6iT - 1.16e13T^{2}
47 1+6.35e6iT−2.38e13T2 1 + 6.35e6iT - 2.38e13T^{2}
53 1−1.06e6T+6.22e13T2 1 - 1.06e6T + 6.22e13T^{2}
59 1+5.76e6iT−1.46e14T2 1 + 5.76e6iT - 1.46e14T^{2}
61 1+1.71e7T+1.91e14T2 1 + 1.71e7T + 1.91e14T^{2}
67 1−2.74e7iT−4.06e14T2 1 - 2.74e7iT - 4.06e14T^{2}
71 1+3.98e7iT−6.45e14T2 1 + 3.98e7iT - 6.45e14T^{2}
73 1+5.32e7T+8.06e14T2 1 + 5.32e7T + 8.06e14T^{2}
79 1−1.82e7iT−1.51e15T2 1 - 1.82e7iT - 1.51e15T^{2}
83 1+7.78e6iT−2.25e15T2 1 + 7.78e6iT - 2.25e15T^{2}
89 1−8.66e7T+3.93e15T2 1 - 8.66e7T + 3.93e15T^{2}
97 1+7.39e7T+7.83e15T2 1 + 7.39e7T + 7.83e15T^{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

−11.17513710424716372152590745335, −9.605679495620223711953035451261, −8.765986195443750263142854088507, −7.58594207577390372571239401753, −7.01299975256885672815583259933, −5.49429930017788776955338228201, −4.46925230223090905293057131511, −2.91902204570390844733044676306, −1.91507540596991583429384839084, −0.13454411129309862285288249384, 0.66558703464493010180718701347, 2.73517508324664461255625713971, 4.06683794083073691805518643085, 4.48370642596426067518237760931, 6.17092911418640255175017106012, 7.48489104635473035342332162386, 8.097004784722189091895479379890, 9.407756429300799375277238845743, 10.50132535522311251840591056381, 11.13402362399870035272037751365

Graph of the ZZ-function along the critical line