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Properties

Label 2-2e4-16.13-c27-0-20
Degree 22
Conductor 1616
Sign 0.853+0.521i0.853 + 0.521i
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  + (−2.70e3 + 1.12e4i)2-s + (−3.16e6 − 3.16e6i)3-s + (−1.19e8 − 6.08e7i)4-s + (−9.16e8 + 9.16e8i)5-s + (4.42e10 − 2.71e10i)6-s − 2.63e11i·7-s + (1.00e12 − 1.18e12i)8-s + 1.24e13i·9-s + (−7.85e12 − 1.28e13i)10-s + (1.27e14 − 1.27e14i)11-s + (1.86e14 + 5.71e14i)12-s + (4.89e13 + 4.89e13i)13-s + (2.96e15 + 7.11e14i)14-s + 5.80e15·15-s + (1.06e16 + 1.45e16i)16-s − 1.67e16·17-s + ⋯
L(s)  = 1  + (−0.233 + 0.972i)2-s + (−1.14 − 1.14i)3-s + (−0.891 − 0.453i)4-s + (−0.335 + 0.335i)5-s + (1.38 − 0.847i)6-s − 1.02i·7-s + (0.648 − 0.761i)8-s + 1.62i·9-s + (−0.248 − 0.404i)10-s + (1.11 − 1.11i)11-s + (0.502 + 1.54i)12-s + (0.0448 + 0.0448i)13-s + (0.999 + 0.239i)14-s + 0.770·15-s + (0.588 + 0.808i)16-s − 0.411·17-s + ⋯

Functional equation

Λ(s)=(16s/2 ΓC(s) L(s)=((0.853+0.521i) Λ‾(28−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(28-s) \end{aligned}
Λ(s)=(16s/2 ΓC(s+27/2) L(s)=((0.853+0.521i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1616    =    242^{4}
Sign: 0.853+0.521i0.853 + 0.521i
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), 0.853+0.521i)(2,\ 16,\ (\ :27/2),\ 0.853 + 0.521i)

Particular Values

L(14)L(14) ≈\approx 0.92617449930.9261744993
L(12)L(\frac12) ≈\approx 0.92617449930.9261744993
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+(2.70e3−1.12e4i)T 1 + (2.70e3 - 1.12e4i)T
good3 1+(3.16e6+3.16e6i)T+7.62e12iT2 1 + (3.16e6 + 3.16e6i)T + 7.62e12iT^{2}
5 1+(9.16e8−9.16e8i)T−7.45e18iT2 1 + (9.16e8 - 9.16e8i)T - 7.45e18iT^{2}
7 1+2.63e11iT−6.57e22T2 1 + 2.63e11iT - 6.57e22T^{2}
11 1+(−1.27e14+1.27e14i)T−1.31e28iT2 1 + (-1.27e14 + 1.27e14i)T - 1.31e28iT^{2}
13 1+(−4.89e13−4.89e13i)T+1.19e30iT2 1 + (-4.89e13 - 4.89e13i)T + 1.19e30iT^{2}
17 1+1.67e16T+1.66e33T2 1 + 1.67e16T + 1.66e33T^{2}
19 1+(1.31e17+1.31e17i)T+3.36e34iT2 1 + (1.31e17 + 1.31e17i)T + 3.36e34iT^{2}
23 1−3.03e18iT−5.84e36T2 1 - 3.03e18iT - 5.84e36T^{2}
29 1+(−5.07e19−5.07e19i)T+3.05e39iT2 1 + (-5.07e19 - 5.07e19i)T + 3.05e39iT^{2}
31 1−1.15e20T+1.84e40T2 1 - 1.15e20T + 1.84e40T^{2}
37 1+(1.72e21−1.72e21i)T−2.19e42iT2 1 + (1.72e21 - 1.72e21i)T - 2.19e42iT^{2}
41 1+4.98e21iT−3.50e43T2 1 + 4.98e21iT - 3.50e43T^{2}
43 1+(−1.24e22+1.24e22i)T−1.26e44iT2 1 + (-1.24e22 + 1.24e22i)T - 1.26e44iT^{2}
47 1−2.25e22T+1.40e45T2 1 - 2.25e22T + 1.40e45T^{2}
53 1+(−1.09e23+1.09e23i)T−3.59e46iT2 1 + (-1.09e23 + 1.09e23i)T - 3.59e46iT^{2}
59 1+(−4.56e23+4.56e23i)T−6.50e47iT2 1 + (-4.56e23 + 4.56e23i)T - 6.50e47iT^{2}
61 1+(−1.24e24−1.24e24i)T+1.59e48iT2 1 + (-1.24e24 - 1.24e24i)T + 1.59e48iT^{2}
67 1+(−1.56e24−1.56e24i)T+2.01e49iT2 1 + (-1.56e24 - 1.56e24i)T + 2.01e49iT^{2}
71 1−3.51e24iT−9.63e49T2 1 - 3.51e24iT - 9.63e49T^{2}
73 1−2.24e25iT−2.04e50T2 1 - 2.24e25iT - 2.04e50T^{2}
79 1+4.59e25T+1.72e51T2 1 + 4.59e25T + 1.72e51T^{2}
83 1+(−4.56e25−4.56e25i)T+6.53e51iT2 1 + (-4.56e25 - 4.56e25i)T + 6.53e51iT^{2}
89 1+2.12e26iT−4.30e52T2 1 + 2.12e26iT - 4.30e52T^{2}
97 1−6.87e26T+4.39e53T2 1 - 6.87e26T + 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

−13.26148154185667407769155624682, −11.66150172616573217318358111371, −10.61234676340945623701353811133, −8.639841076205709328532987886368, −7.12191107527070477830316878000, −6.72016396187780881171999186085, −5.46675042132031623912795765353, −3.88715036398401306413961376906, −1.24476255522736685340585146496, −0.56308308266378782693149875208, 0.63675391437610185963414441004, 2.26701798500367960980060987519, 4.12468263878323037458248734119, 4.66164421518737569439066729525, 6.16903874714335691100423018299, 8.578873388861248349733317199197, 9.648104022119705013817284908266, 10.66067216875668274899673370655, 11.99414076220771820109827826391, 12.31189990327391707802023853376

Graph of the ZZ-function along the critical line