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Properties

Label 2-2e4-16.13-c27-0-27
Degree 22
Conductor 1616
Sign 0.971+0.235i0.971 + 0.235i
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  + (−1.14e4 − 1.81e3i)2-s + (1.53e5 + 1.53e5i)3-s + (1.27e8 + 4.15e7i)4-s + (2.87e9 − 2.87e9i)5-s + (−1.47e9 − 2.03e9i)6-s + 4.55e11i·7-s + (−1.38e12 − 7.07e11i)8-s − 7.57e12i·9-s + (−3.80e13 + 2.76e13i)10-s + (2.18e13 − 2.18e13i)11-s + (1.32e13 + 2.60e13i)12-s + (6.58e13 + 6.58e13i)13-s + (8.27e14 − 5.20e15i)14-s + 8.82e14·15-s + (1.45e16 + 1.06e16i)16-s + 3.33e16·17-s + ⋯
L(s)  = 1  + (−0.987 − 0.156i)2-s + (0.0556 + 0.0556i)3-s + (0.950 + 0.309i)4-s + (1.05 − 1.05i)5-s + (−0.0462 − 0.0636i)6-s + 1.77i·7-s + (−0.890 − 0.455i)8-s − 0.993i·9-s + (−1.20 + 0.873i)10-s + (0.190 − 0.190i)11-s + (0.0356 + 0.0701i)12-s + (0.0603 + 0.0603i)13-s + (0.278 − 1.75i)14-s + 0.117·15-s + (0.808 + 0.589i)16-s + 0.816·17-s + ⋯

Functional equation

Λ(s)=(16s/2 ΓC(s) L(s)=((0.971+0.235i) Λ‾(28−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(28-s) \end{aligned}
Λ(s)=(16s/2 ΓC(s+27/2) L(s)=((0.971+0.235i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1616    =    242^{4}
Sign: 0.971+0.235i0.971 + 0.235i
Analytic conductor: 73.896873.8968
Root analytic conductor: 8.596338.59633
Motivic weight: 2727
Rational: no
Arithmetic: yes
Character: χ16(13,⋅)\chi_{16} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 16, ( :27/2), 0.971+0.235i)(2,\ 16,\ (\ :27/2),\ 0.971 + 0.235i)

Particular Values

L(14)L(14) ≈\approx 1.9092717701.909271770
L(12)L(\frac12) ≈\approx 1.9092717701.909271770
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+(1.14e4+1.81e3i)T 1 + (1.14e4 + 1.81e3i)T
good3 1+(−1.53e5−1.53e5i)T+7.62e12iT2 1 + (-1.53e5 - 1.53e5i)T + 7.62e12iT^{2}
5 1+(−2.87e9+2.87e9i)T−7.45e18iT2 1 + (-2.87e9 + 2.87e9i)T - 7.45e18iT^{2}
7 1−4.55e11iT−6.57e22T2 1 - 4.55e11iT - 6.57e22T^{2}
11 1+(−2.18e13+2.18e13i)T−1.31e28iT2 1 + (-2.18e13 + 2.18e13i)T - 1.31e28iT^{2}
13 1+(−6.58e13−6.58e13i)T+1.19e30iT2 1 + (-6.58e13 - 6.58e13i)T + 1.19e30iT^{2}
17 1−3.33e16T+1.66e33T2 1 - 3.33e16T + 1.66e33T^{2}
19 1+(−2.22e17−2.22e17i)T+3.36e34iT2 1 + (-2.22e17 - 2.22e17i)T + 3.36e34iT^{2}
23 1+4.14e18iT−5.84e36T2 1 + 4.14e18iT - 5.84e36T^{2}
29 1+(−1.87e19−1.87e19i)T+3.05e39iT2 1 + (-1.87e19 - 1.87e19i)T + 3.05e39iT^{2}
31 1+5.35e19T+1.84e40T2 1 + 5.35e19T + 1.84e40T^{2}
37 1+(1.64e21−1.64e21i)T−2.19e42iT2 1 + (1.64e21 - 1.64e21i)T - 2.19e42iT^{2}
41 1−3.79e21iT−3.50e43T2 1 - 3.79e21iT - 3.50e43T^{2}
43 1+(−4.06e21+4.06e21i)T−1.26e44iT2 1 + (-4.06e21 + 4.06e21i)T - 1.26e44iT^{2}
47 1−3.33e22T+1.40e45T2 1 - 3.33e22T + 1.40e45T^{2}
53 1+(7.44e22−7.44e22i)T−3.59e46iT2 1 + (7.44e22 - 7.44e22i)T - 3.59e46iT^{2}
59 1+(−3.85e23+3.85e23i)T−6.50e47iT2 1 + (-3.85e23 + 3.85e23i)T - 6.50e47iT^{2}
61 1+(−3.77e23−3.77e23i)T+1.59e48iT2 1 + (-3.77e23 - 3.77e23i)T + 1.59e48iT^{2}
67 1+(−1.29e24−1.29e24i)T+2.01e49iT2 1 + (-1.29e24 - 1.29e24i)T + 2.01e49iT^{2}
71 1+1.59e25iT−9.63e49T2 1 + 1.59e25iT - 9.63e49T^{2}
73 1−1.45e25iT−2.04e50T2 1 - 1.45e25iT - 2.04e50T^{2}
79 1−3.96e25T+1.72e51T2 1 - 3.96e25T + 1.72e51T^{2}
83 1+(2.74e25+2.74e25i)T+6.53e51iT2 1 + (2.74e25 + 2.74e25i)T + 6.53e51iT^{2}
89 1−1.01e26iT−4.30e52T2 1 - 1.01e26iT - 4.30e52T^{2}
97 1−5.84e26T+4.39e53T2 1 - 5.84e26T + 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

−12.44283407546434527907263664617, −12.04196940716129625202403193662, −9.955923661993306853621185831382, −9.117982495180216329296650984198, −8.406932819724664413845543673308, −6.27673517944028928233801501935, −5.44110128094974632395916508806, −3.09353767280761871271290337562, −1.82755633013662156051619433355, −0.865206185336428637119140300180, 0.850575904183178792192576853660, 1.94876025603849599209867784113, 3.29760853300294552866258131228, 5.48347884687698229228726370071, 7.06165303233027603794112528232, 7.51259314933724282069589425105, 9.575790436158647142431013915554, 10.39426666181940888494137017447, 11.19501263564207131832419086566, 13.60080845158248424448084788127

Graph of the ZZ-function along the critical line