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Properties

Label 2-2e3-8.3-c18-0-5
Degree 22
Conductor 88
Sign 0.959+0.283i0.959 + 0.283i
Analytic cond. 16.430816.4308
Root an. cond. 4.053504.05350
Motivic weight 1818
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (297. − 416. i)2-s − 2.59e4·3-s + (−8.54e4 − 2.47e5i)4-s + 1.90e6i·5-s + (−7.70e6 + 1.08e7i)6-s + 9.33e6i·7-s + (−1.28e8 − 3.80e7i)8-s + 2.84e8·9-s + (7.93e8 + 5.65e8i)10-s + 2.94e9·11-s + (2.21e9 + 6.42e9i)12-s − 7.67e9i·13-s + (3.89e9 + 2.77e9i)14-s − 4.93e10i·15-s + (−5.41e10 + 4.23e10i)16-s − 1.40e10·17-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)2-s − 1.31·3-s + (−0.326 − 0.945i)4-s + 0.974i·5-s + (−0.764 + 1.07i)6-s + 0.231i·7-s + (−0.959 − 0.283i)8-s + 0.734·9-s + (0.793 + 0.565i)10-s + 1.24·11-s + (0.429 + 1.24i)12-s − 0.723i·13-s + (0.188 + 0.134i)14-s − 1.28i·15-s + (−0.787 + 0.616i)16-s − 0.118·17-s + ⋯

Functional equation

Λ(s)=(8s/2 ΓC(s) L(s)=((0.959+0.283i) Λ‾(19−s)\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(19-s) \end{aligned}
Λ(s)=(8s/2 ΓC(s+9) L(s)=((0.959+0.283i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 88    =    232^{3}
Sign: 0.959+0.283i0.959 + 0.283i
Analytic conductor: 16.430816.4308
Root analytic conductor: 4.053504.05350
Motivic weight: 1818
Rational: no
Arithmetic: yes
Character: χ8(3,⋅)\chi_{8} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 8, ( :9), 0.959+0.283i)(2,\ 8,\ (\ :9),\ 0.959 + 0.283i)

Particular Values

L(192)L(\frac{19}{2}) ≈\approx 1.45521−0.210395i1.45521 - 0.210395i
L(12)L(\frac12) ≈\approx 1.45521−0.210395i1.45521 - 0.210395i
L(10)L(10) 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+(−297.+416.i)T 1 + (-297. + 416. i)T
good3 1+2.59e4T+3.87e8T2 1 + 2.59e4T + 3.87e8T^{2}
5 1−1.90e6iT−3.81e12T2 1 - 1.90e6iT - 3.81e12T^{2}
7 1−9.33e6iT−1.62e15T2 1 - 9.33e6iT - 1.62e15T^{2}
11 1−2.94e9T+5.55e18T2 1 - 2.94e9T + 5.55e18T^{2}
13 1+7.67e9iT−1.12e20T2 1 + 7.67e9iT - 1.12e20T^{2}
17 1+1.40e10T+1.40e22T2 1 + 1.40e10T + 1.40e22T^{2}
19 1−5.59e11T+1.04e23T2 1 - 5.59e11T + 1.04e23T^{2}
23 1−2.61e12iT−3.24e24T2 1 - 2.61e12iT - 3.24e24T^{2}
29 1+1.91e13iT−2.10e26T2 1 + 1.91e13iT - 2.10e26T^{2}
31 1−3.57e13iT−6.99e26T2 1 - 3.57e13iT - 6.99e26T^{2}
37 1−2.51e14iT−1.68e28T2 1 - 2.51e14iT - 1.68e28T^{2}
41 1−7.94e13T+1.07e29T2 1 - 7.94e13T + 1.07e29T^{2}
43 1+3.62e14T+2.52e29T2 1 + 3.62e14T + 2.52e29T^{2}
47 1+2.20e14iT−1.25e30T2 1 + 2.20e14iT - 1.25e30T^{2}
53 1−1.22e15iT−1.08e31T2 1 - 1.22e15iT - 1.08e31T^{2}
59 1−2.66e15T+7.50e31T2 1 - 2.66e15T + 7.50e31T^{2}
61 1+2.02e16iT−1.36e32T2 1 + 2.02e16iT - 1.36e32T^{2}
67 1−3.97e16T+7.40e32T2 1 - 3.97e16T + 7.40e32T^{2}
71 1+1.25e16iT−2.10e33T2 1 + 1.25e16iT - 2.10e33T^{2}
73 1−6.29e15T+3.46e33T2 1 - 6.29e15T + 3.46e33T^{2}
79 1−6.53e16iT−1.43e34T2 1 - 6.53e16iT - 1.43e34T^{2}
83 1−1.09e17T+3.49e34T2 1 - 1.09e17T + 3.49e34T^{2}
89 1+3.14e17T+1.22e35T2 1 + 3.14e17T + 1.22e35T^{2}
97 1+5.54e17T+5.77e35T2 1 + 5.54e17T + 5.77e35T^{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

−17.44298608354067781236448293447, −15.43748065701116826457016989369, −13.91165048342826418290050171449, −11.99608350902543324609138878767, −11.26796881720380996906952701614, −9.870160987630335420270923633952, −6.57595996302880613573781802077, −5.29345763252019602704798081606, −3.28423586349785818691952250609, −1.08211383900967262149149125038, 0.74656490240077203209065515775, 4.25968940074835591751080093010, 5.50382863404481145020222381099, 6.89269041380690275825763022934, 9.028670039137924168588751575557, 11.57187972950771222353969881416, 12.55794038138931590542901979668, 14.25417141907341205119005754918, 16.35806505623895132148519048638, 16.72091730765816021787088908069

Graph of the ZZ-function along the critical line