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Properties

Label 2-2e4-16.5-c27-0-9
Degree 22
Conductor 1616
Sign 0.959+0.283i0.959 + 0.283i
Analytic cond. 73.896873.8968
Root an. cond. 8.596338.59633
Motivic weight 2727
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.15e4 + 305. i)2-s + (−2.33e6 + 2.33e6i)3-s + (1.34e8 − 7.08e6i)4-s + (−3.25e9 − 3.25e9i)5-s + (2.62e10 − 2.77e10i)6-s − 4.57e11i·7-s + (−1.55e12 + 1.23e11i)8-s − 3.25e12i·9-s + (3.86e13 + 3.67e13i)10-s + (7.31e13 + 7.31e13i)11-s + (−2.96e14 + 3.29e14i)12-s + (−1.05e15 + 1.05e15i)13-s + (1.40e14 + 5.30e15i)14-s + 1.51e16·15-s + (1.79e16 − 1.89e15i)16-s − 3.52e16·17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0264i)2-s + (−0.844 + 0.844i)3-s + (0.998 − 0.0527i)4-s + (−1.19 − 1.19i)5-s + (0.822 − 0.866i)6-s − 1.78i·7-s + (−0.996 + 0.0791i)8-s − 0.426i·9-s + (1.22 + 1.16i)10-s + (0.638 + 0.638i)11-s + (−0.798 + 0.888i)12-s + (−0.962 + 0.962i)13-s + (0.0471 + 1.78i)14-s + 2.01·15-s + (0.994 − 0.105i)16-s − 0.863·17-s + ⋯

Functional equation

Λ(s)=(16s/2 ΓC(s) L(s)=((0.959+0.283i) Λ‾(28−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(28-s) \end{aligned}
Λ(s)=(16s/2 ΓC(s+27/2) L(s)=((0.959+0.283i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1616    =    242^{4}
Sign: 0.959+0.283i0.959 + 0.283i
Analytic conductor: 73.896873.8968
Root analytic conductor: 8.596338.59633
Motivic weight: 2727
Rational: no
Arithmetic: yes
Character: χ16(5,⋅)\chi_{16} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 16, ( :27/2), 0.959+0.283i)(2,\ 16,\ (\ :27/2),\ 0.959 + 0.283i)

Particular Values

L(14)L(14) ≈\approx 0.18820060140.1882006014
L(12)L(\frac12) ≈\approx 0.18820060140.1882006014
L(292)L(\frac{29}{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+(1.15e4−305.i)T 1 + (1.15e4 - 305. i)T
good3 1+(2.33e6−2.33e6i)T−7.62e12iT2 1 + (2.33e6 - 2.33e6i)T - 7.62e12iT^{2}
5 1+(3.25e9+3.25e9i)T+7.45e18iT2 1 + (3.25e9 + 3.25e9i)T + 7.45e18iT^{2}
7 1+4.57e11iT−6.57e22T2 1 + 4.57e11iT - 6.57e22T^{2}
11 1+(−7.31e13−7.31e13i)T+1.31e28iT2 1 + (-7.31e13 - 7.31e13i)T + 1.31e28iT^{2}
13 1+(1.05e15−1.05e15i)T−1.19e30iT2 1 + (1.05e15 - 1.05e15i)T - 1.19e30iT^{2}
17 1+3.52e16T+1.66e33T2 1 + 3.52e16T + 1.66e33T^{2}
19 1+(1.46e17−1.46e17i)T−3.36e34iT2 1 + (1.46e17 - 1.46e17i)T - 3.36e34iT^{2}
23 1+2.95e18iT−5.84e36T2 1 + 2.95e18iT - 5.84e36T^{2}
29 1+(3.27e19−3.27e19i)T−3.05e39iT2 1 + (3.27e19 - 3.27e19i)T - 3.05e39iT^{2}
31 1+7.52e18T+1.84e40T2 1 + 7.52e18T + 1.84e40T^{2}
37 1+(2.52e20+2.52e20i)T+2.19e42iT2 1 + (2.52e20 + 2.52e20i)T + 2.19e42iT^{2}
41 1−3.98e21iT−3.50e43T2 1 - 3.98e21iT - 3.50e43T^{2}
43 1+(−1.03e22−1.03e22i)T+1.26e44iT2 1 + (-1.03e22 - 1.03e22i)T + 1.26e44iT^{2}
47 1+2.78e22T+1.40e45T2 1 + 2.78e22T + 1.40e45T^{2}
53 1+(3.83e21+3.83e21i)T+3.59e46iT2 1 + (3.83e21 + 3.83e21i)T + 3.59e46iT^{2}
59 1+(5.02e23+5.02e23i)T+6.50e47iT2 1 + (5.02e23 + 5.02e23i)T + 6.50e47iT^{2}
61 1+(1.45e24−1.45e24i)T−1.59e48iT2 1 + (1.45e24 - 1.45e24i)T - 1.59e48iT^{2}
67 1+(3.20e24−3.20e24i)T−2.01e49iT2 1 + (3.20e24 - 3.20e24i)T - 2.01e49iT^{2}
71 1−2.21e24iT−9.63e49T2 1 - 2.21e24iT - 9.63e49T^{2}
73 1+8.58e24iT−2.04e50T2 1 + 8.58e24iT - 2.04e50T^{2}
79 1−2.53e25T+1.72e51T2 1 - 2.53e25T + 1.72e51T^{2}
83 1+(4.74e25−4.74e25i)T−6.53e51iT2 1 + (4.74e25 - 4.74e25i)T - 6.53e51iT^{2}
89 1+7.83e25iT−4.30e52T2 1 + 7.83e25iT - 4.30e52T^{2}
97 1+7.63e26T+4.39e53T2 1 + 7.63e26T + 4.39e53T^{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

−12.50619780979329794711634531148, −11.40338182265039640239719724107, −10.46664465519401972903965033134, −9.281659758834378174727742466825, −7.84216501236718746561661029171, −6.75175119584775595745174149304, −4.54819389286975199719249383992, −4.14834797819416378377998876988, −1.48390514784318482086041793254, −0.25104264059367360516441378788, 0.28873841765962768660857300375, 2.10221468058390167469479308397, 3.14749591947077518739565059629, 5.78121145912201627209838685157, 6.72870912013469932812011328265, 7.76089135811001065754497909522, 9.082062564590294671818408537813, 10.99350180190961756847165892795, 11.65990684705579430198211076366, 12.42591968720588945767893030576

Graph of the ZZ-function along the critical line