↑

Properties

Label 2-22-11.5-c13-0-6
Degree 22
Conductor 2222
Sign 0.515−0.856i0.515 - 0.856i
Analytic cond. 23.590823.5908
Root an. cond. 4.857034.85703
Motivic weight 1313
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (19.7 + 60.8i)2-s + (−306. − 223. i)3-s + (−3.31e3 + 2.40e3i)4-s + (−5.17e3 + 1.59e4i)5-s + (7.50e3 − 2.30e4i)6-s + (2.91e5 − 2.11e5i)7-s + (−2.12e5 − 1.54e5i)8-s + (−4.48e5 − 1.37e6i)9-s − 1.07e6·10-s + (2.53e6 + 5.29e6i)11-s + 1.55e6·12-s + (−2.24e6 − 6.91e6i)13-s + (1.86e7 + 1.35e7i)14-s + (5.14e6 − 3.73e6i)15-s + (5.18e6 − 1.59e7i)16-s + (7.19e6 − 2.21e7i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.243 − 0.176i)3-s + (−0.404 + 0.293i)4-s + (−0.148 + 0.456i)5-s + (0.0656 − 0.202i)6-s + (0.936 − 0.680i)7-s + (−0.286 − 0.207i)8-s + (−0.281 − 0.865i)9-s − 0.339·10-s + (0.431 + 0.901i)11-s + 0.150·12-s + (−0.129 − 0.397i)13-s + (0.662 + 0.481i)14-s + (0.116 − 0.0847i)15-s + (0.0772 − 0.237i)16-s + (0.0723 − 0.222i)17-s + ⋯

Functional equation

Λ(s)=(22s/2 ΓC(s) L(s)=((0.515−0.856i) Λ‾(14−s)\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(14-s) \end{aligned}
Λ(s)=(22s/2 ΓC(s+13/2) L(s)=((0.515−0.856i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2222    =    2⋅112 \cdot 11
Sign: 0.515−0.856i0.515 - 0.856i
Analytic conductor: 23.590823.5908
Root analytic conductor: 4.857034.85703
Motivic weight: 1313
Rational: no
Arithmetic: yes
Character: χ22(5,⋅)\chi_{22} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 22, ( :13/2), 0.515−0.856i)(2,\ 22,\ (\ :13/2),\ 0.515 - 0.856i)

Particular Values

L(7)L(7) ≈\approx 2.0632940732.063294073
L(12)L(\frac12) ≈\approx 2.0632940732.063294073
L(152)L(\frac{15}{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+(−19.7−60.8i)T 1 + (-19.7 - 60.8i)T
11 1+(−2.53e6−5.29e6i)T 1 + (-2.53e6 - 5.29e6i)T
good3 1+(306.+223.i)T+(4.92e5+1.51e6i)T2 1 + (306. + 223. i)T + (4.92e5 + 1.51e6i)T^{2}
5 1+(5.17e3−1.59e4i)T+(−9.87e8−7.17e8i)T2 1 + (5.17e3 - 1.59e4i)T + (-9.87e8 - 7.17e8i)T^{2}
7 1+(−2.91e5+2.11e5i)T+(2.99e10−9.21e10i)T2 1 + (-2.91e5 + 2.11e5i)T + (2.99e10 - 9.21e10i)T^{2}
13 1+(2.24e6+6.91e6i)T+(−2.45e14+1.78e14i)T2 1 + (2.24e6 + 6.91e6i)T + (-2.45e14 + 1.78e14i)T^{2}
17 1+(−7.19e6+2.21e7i)T+(−8.01e15−5.82e15i)T2 1 + (-7.19e6 + 2.21e7i)T + (-8.01e15 - 5.82e15i)T^{2}
19 1+(−3.25e8−2.36e8i)T+(1.29e16+3.99e16i)T2 1 + (-3.25e8 - 2.36e8i)T + (1.29e16 + 3.99e16i)T^{2}
23 1−2.99e7T+5.04e17T2 1 - 2.99e7T + 5.04e17T^{2}
29 1+(1.85e9−1.34e9i)T+(3.17e18−9.75e18i)T2 1 + (1.85e9 - 1.34e9i)T + (3.17e18 - 9.75e18i)T^{2}
31 1+(−2.30e9−7.08e9i)T+(−1.97e19+1.43e19i)T2 1 + (-2.30e9 - 7.08e9i)T + (-1.97e19 + 1.43e19i)T^{2}
37 1+(−1.67e10+1.21e10i)T+(7.52e19−2.31e20i)T2 1 + (-1.67e10 + 1.21e10i)T + (7.52e19 - 2.31e20i)T^{2}
41 1+(−3.86e10−2.80e10i)T+(2.85e20+8.79e20i)T2 1 + (-3.86e10 - 2.80e10i)T + (2.85e20 + 8.79e20i)T^{2}
43 1−8.85e9T+1.71e21T2 1 - 8.85e9T + 1.71e21T^{2}
47 1+(9.63e10+7.00e10i)T+(1.68e21+5.19e21i)T2 1 + (9.63e10 + 7.00e10i)T + (1.68e21 + 5.19e21i)T^{2}
53 1+(5.59e10+1.72e11i)T+(−2.10e22+1.53e22i)T2 1 + (5.59e10 + 1.72e11i)T + (-2.10e22 + 1.53e22i)T^{2}
59 1+(−4.29e11+3.11e11i)T+(3.24e22−9.98e22i)T2 1 + (-4.29e11 + 3.11e11i)T + (3.24e22 - 9.98e22i)T^{2}
61 1+(2.13e10−6.57e10i)T+(−1.30e23−9.51e22i)T2 1 + (2.13e10 - 6.57e10i)T + (-1.30e23 - 9.51e22i)T^{2}
67 1−6.63e11T+5.48e23T2 1 - 6.63e11T + 5.48e23T^{2}
71 1+(−4.27e11+1.31e12i)T+(−9.42e23−6.84e23i)T2 1 + (-4.27e11 + 1.31e12i)T + (-9.42e23 - 6.84e23i)T^{2}
73 1+(−5.16e11+3.74e11i)T+(5.16e23−1.59e24i)T2 1 + (-5.16e11 + 3.74e11i)T + (5.16e23 - 1.59e24i)T^{2}
79 1+(−9.73e11−2.99e12i)T+(−3.77e24+2.74e24i)T2 1 + (-9.73e11 - 2.99e12i)T + (-3.77e24 + 2.74e24i)T^{2}
83 1+(9.67e10−2.97e11i)T+(−7.17e24−5.21e24i)T2 1 + (9.67e10 - 2.97e11i)T + (-7.17e24 - 5.21e24i)T^{2}
89 1+1.99e11T+2.19e25T2 1 + 1.99e11T + 2.19e25T^{2}
97 1+(−1.77e12−5.47e12i)T+(−5.44e25+3.95e25i)T2 1 + (-1.77e12 - 5.47e12i)T + (-5.44e25 + 3.95e25i)T^{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

−14.80636144111332575292303666812, −14.29225249099505968915300235757, −12.54499197932299064111941793042, −11.33039803122691573836915172633, −9.644860479923519511759146539855, −7.83305585812215289309824635657, −6.80795869623720756254488245886, −5.16108809553090833352078999671, −3.55258055173388150933075859145, −1.10335669890447671765798680410, 0.906284489657855779926924369229, 2.53880569585503031971410334308, 4.52324291826598085428187647812, 5.62901622952078490082015341445, 8.072746160415981620693785629952, 9.329738336201670698760203963774, 11.17191135827224048227350032614, 11.70757503872196432747286328641, 13.35606783644232897747500943013, 14.46464501207019662475115631103

Graph of the ZZ-function along the critical line