↑

Properties

Label 2-2e3-8.5-c19-0-17
Degree 22
Conductor 88
Sign −0.440−0.897i-0.440 - 0.897i
Analytic cond. 18.305318.3053
Root an. cond. 4.278474.27847
Motivic weight 1919
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (564. − 452. i)2-s − 5.04e4i·3-s + (1.14e5 − 5.11e5i)4-s + 7.26e6i·5-s + (−2.28e7 − 2.84e7i)6-s − 1.80e8·7-s + (−1.67e8 − 3.40e8i)8-s − 1.38e9·9-s + (3.28e9 + 4.10e9i)10-s + 1.74e9i·11-s + (−2.58e10 − 5.75e9i)12-s + 9.15e9i·13-s + (−1.02e11 + 8.19e10i)14-s + 3.66e11·15-s + (−2.48e11 − 1.16e11i)16-s − 2.51e11·17-s + ⋯
L(s)  = 1  + (0.780 − 0.625i)2-s − 1.47i·3-s + (0.217 − 0.976i)4-s + 1.66i·5-s + (−0.925 − 1.15i)6-s − 1.69·7-s + (−0.440 − 0.897i)8-s − 1.18·9-s + (1.04 + 1.29i)10-s + 0.223i·11-s + (−1.44 − 0.321i)12-s + 0.239i·13-s + (−1.32 + 1.06i)14-s + 2.46·15-s + (−0.905 − 0.424i)16-s − 0.514·17-s + ⋯

Functional equation

Λ(s)=(8s/2 ΓC(s) L(s)=((−0.440−0.897i) Λ‾(20−s)\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(20-s) \end{aligned}
Λ(s)=(8s/2 ΓC(s+19/2) L(s)=((−0.440−0.897i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 88    =    232^{3}
Sign: −0.440−0.897i-0.440 - 0.897i
Analytic conductor: 18.305318.3053
Root analytic conductor: 4.278474.27847
Motivic weight: 1919
Rational: no
Arithmetic: yes
Character: χ8(5,⋅)\chi_{8} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 8, ( :19/2), −0.440−0.897i)(2,\ 8,\ (\ :19/2),\ -0.440 - 0.897i)

Particular Values

L(10)L(10) ≈\approx 0.315465+0.506374i0.315465 + 0.506374i
L(12)L(\frac12) ≈\approx 0.315465+0.506374i0.315465 + 0.506374i
L(212)L(\frac{21}{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+(−564.+452.i)T 1 + (-564. + 452. i)T
good3 1+5.04e4iT−1.16e9T2 1 + 5.04e4iT - 1.16e9T^{2}
5 1−7.26e6iT−1.90e13T2 1 - 7.26e6iT - 1.90e13T^{2}
7 1+1.80e8T+1.13e16T2 1 + 1.80e8T + 1.13e16T^{2}
11 1−1.74e9iT−6.11e19T2 1 - 1.74e9iT - 6.11e19T^{2}
13 1−9.15e9iT−1.46e21T2 1 - 9.15e9iT - 1.46e21T^{2}
17 1+2.51e11T+2.39e23T2 1 + 2.51e11T + 2.39e23T^{2}
19 1+1.49e12iT−1.97e24T2 1 + 1.49e12iT - 1.97e24T^{2}
23 1+4.50e12T+7.46e25T2 1 + 4.50e12T + 7.46e25T^{2}
29 1−3.77e12iT−6.10e27T2 1 - 3.77e12iT - 6.10e27T^{2}
31 1−1.61e14T+2.16e28T2 1 - 1.61e14T + 2.16e28T^{2}
37 1−4.58e13iT−6.24e29T2 1 - 4.58e13iT - 6.24e29T^{2}
41 1+3.30e15T+4.39e30T2 1 + 3.30e15T + 4.39e30T^{2}
43 1+7.16e14iT−1.08e31T2 1 + 7.16e14iT - 1.08e31T^{2}
47 1+9.61e15T+5.88e31T2 1 + 9.61e15T + 5.88e31T^{2}
53 1+2.87e16iT−5.77e32T2 1 + 2.87e16iT - 5.77e32T^{2}
59 1+8.42e16iT−4.42e33T2 1 + 8.42e16iT - 4.42e33T^{2}
61 1−3.85e16iT−8.34e33T2 1 - 3.85e16iT - 8.34e33T^{2}
67 1−3.60e16iT−4.95e34T2 1 - 3.60e16iT - 4.95e34T^{2}
71 1+8.39e16T+1.49e35T2 1 + 8.39e16T + 1.49e35T^{2}
73 1−1.83e14T+2.53e35T2 1 - 1.83e14T + 2.53e35T^{2}
79 1−1.00e18T+1.13e36T2 1 - 1.00e18T + 1.13e36T^{2}
83 1−1.33e18iT−2.90e36T2 1 - 1.33e18iT - 2.90e36T^{2}
89 1+4.03e18T+1.09e37T2 1 + 4.03e18T + 1.09e37T^{2}
97 1−6.57e18T+5.60e37T2 1 - 6.57e18T + 5.60e37T^{2}
show more
show less
   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

−15.45169914520973652218844240165, −13.88351288117272109138450472817, −12.99592600757690499777399445122, −11.61026403262032826040247729269, −10.02003464351836960419835104973, −6.82155267291466347723436271281, −6.46630035094955359970668581403, −3.23647478322546103715717056289, −2.24207716039105267176272163400, −0.15850747222322163598465458566, 3.45213717485655731975946873263, 4.59565096295353086229317582821, 5.93340855926478710327025434777, 8.608055511508073105083978559096, 9.837339334926724416030726418872, 12.25201770360627609552907192627, 13.42932994656456753519936081284, 15.47120025804113693228605978061, 16.25319614665928360176651979395, 16.84752622187531768493383007353

Graph of the ZZ-function along the critical line