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Properties

Label 2-2e4-16.5-c27-0-49
Degree 22
Conductor 1616
Sign −0.976+0.217i-0.976 + 0.217i
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  + (−6.84e3 + 9.34e3i)2-s + (2.95e6 − 2.95e6i)3-s + (−4.04e7 − 1.27e8i)4-s + (6.75e8 + 6.75e8i)5-s + (7.38e9 + 4.78e10i)6-s − 3.29e11i·7-s + (1.47e12 + 4.98e11i)8-s − 9.87e12i·9-s + (−1.09e13 + 1.68e12i)10-s + (−1.51e13 − 1.51e13i)11-s + (−4.98e14 − 2.58e14i)12-s + (−4.13e14 + 4.13e14i)13-s + (3.08e15 + 2.25e15i)14-s + 3.99e15·15-s + (−1.47e16 + 1.03e16i)16-s − 6.20e16·17-s + ⋯
L(s)  = 1  + (−0.591 + 0.806i)2-s + (1.07 − 1.07i)3-s + (−0.301 − 0.953i)4-s + (0.247 + 0.247i)5-s + (0.230 + 1.49i)6-s − 1.28i·7-s + (0.947 + 0.320i)8-s − 1.29i·9-s + (−0.346 + 0.0533i)10-s + (−0.132 − 0.132i)11-s + (−1.34 − 0.698i)12-s + (−0.378 + 0.378i)13-s + (1.03 + 0.760i)14-s + 0.530·15-s + (−0.818 + 0.574i)16-s − 1.51·17-s + ⋯

Functional equation

Λ(s)=(16s/2 ΓC(s) L(s)=((−0.976+0.217i) Λ‾(28−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(28-s) \end{aligned}
Λ(s)=(16s/2 ΓC(s+27/2) L(s)=((−0.976+0.217i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1616    =    242^{4}
Sign: −0.976+0.217i-0.976 + 0.217i
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), −0.976+0.217i)(2,\ 16,\ (\ :27/2),\ -0.976 + 0.217i)

Particular Values

L(14)L(14) ≈\approx 1.0512145381.051214538
L(12)L(\frac12) ≈\approx 1.0512145381.051214538
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+(6.84e3−9.34e3i)T 1 + (6.84e3 - 9.34e3i)T
good3 1+(−2.95e6+2.95e6i)T−7.62e12iT2 1 + (-2.95e6 + 2.95e6i)T - 7.62e12iT^{2}
5 1+(−6.75e8−6.75e8i)T+7.45e18iT2 1 + (-6.75e8 - 6.75e8i)T + 7.45e18iT^{2}
7 1+3.29e11iT−6.57e22T2 1 + 3.29e11iT - 6.57e22T^{2}
11 1+(1.51e13+1.51e13i)T+1.31e28iT2 1 + (1.51e13 + 1.51e13i)T + 1.31e28iT^{2}
13 1+(4.13e14−4.13e14i)T−1.19e30iT2 1 + (4.13e14 - 4.13e14i)T - 1.19e30iT^{2}
17 1+6.20e16T+1.66e33T2 1 + 6.20e16T + 1.66e33T^{2}
19 1+(−1.52e17+1.52e17i)T−3.36e34iT2 1 + (-1.52e17 + 1.52e17i)T - 3.36e34iT^{2}
23 1+2.08e18iT−5.84e36T2 1 + 2.08e18iT - 5.84e36T^{2}
29 1+(1.58e19−1.58e19i)T−3.05e39iT2 1 + (1.58e19 - 1.58e19i)T - 3.05e39iT^{2}
31 1+1.09e20T+1.84e40T2 1 + 1.09e20T + 1.84e40T^{2}
37 1+(−7.19e20−7.19e20i)T+2.19e42iT2 1 + (-7.19e20 - 7.19e20i)T + 2.19e42iT^{2}
41 1−8.61e21iT−3.50e43T2 1 - 8.61e21iT - 3.50e43T^{2}
43 1+(−1.75e21−1.75e21i)T+1.26e44iT2 1 + (-1.75e21 - 1.75e21i)T + 1.26e44iT^{2}
47 1−1.47e22T+1.40e45T2 1 - 1.47e22T + 1.40e45T^{2}
53 1+(2.01e23+2.01e23i)T+3.59e46iT2 1 + (2.01e23 + 2.01e23i)T + 3.59e46iT^{2}
59 1+(−9.87e22−9.87e22i)T+6.50e47iT2 1 + (-9.87e22 - 9.87e22i)T + 6.50e47iT^{2}
61 1+(9.38e23−9.38e23i)T−1.59e48iT2 1 + (9.38e23 - 9.38e23i)T - 1.59e48iT^{2}
67 1+(2.27e24−2.27e24i)T−2.01e49iT2 1 + (2.27e24 - 2.27e24i)T - 2.01e49iT^{2}
71 1+9.77e24iT−9.63e49T2 1 + 9.77e24iT - 9.63e49T^{2}
73 1−1.86e25iT−2.04e50T2 1 - 1.86e25iT - 2.04e50T^{2}
79 1−6.34e25T+1.72e51T2 1 - 6.34e25T + 1.72e51T^{2}
83 1+(7.33e25−7.33e25i)T−6.53e51iT2 1 + (7.33e25 - 7.33e25i)T - 6.53e51iT^{2}
89 1+3.71e26iT−4.30e52T2 1 + 3.71e26iT - 4.30e52T^{2}
97 1+2.24e26T+4.39e53T2 1 + 2.24e26T + 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

−13.19058048790582830387736299082, −10.91595797141228816666297575760, −9.481895978366492302383172897958, −8.293762817392725592149478336038, −7.20976543332528418700932656818, −6.59441407396138371219969352305, −4.48209474878273584713207372716, −2.58403065864523778335745081155, −1.37045908619897524653307093433, −0.24820173663156585387341877960, 1.85010374533680320129993083067, 2.74811792501876156401666587116, 3.86249718336451673249110985517, 5.27223966975023979359973382337, 7.80202490505116719541922113908, 9.116128401962582296322268033262, 9.384130606612039579463463760718, 10.83733565153397886868899581435, 12.28685940210181516974726991559, 13.61556909059887025212363532236

Graph of the ZZ-function along the critical line