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Properties

Label 2-2e4-16.13-c27-0-12
Degree 22
Conductor 1616
Sign 0.564+0.825i0.564 + 0.825i
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  + (3.87e3 − 1.09e4i)2-s + (−2.14e6 − 2.14e6i)3-s + (−1.04e8 − 8.45e7i)4-s + (1.53e9 − 1.53e9i)5-s + (−3.16e10 + 1.50e10i)6-s + 9.29e10i·7-s + (−1.32e12 + 8.10e11i)8-s + 1.55e12i·9-s + (−1.08e13 − 2.27e13i)10-s + (−6.85e13 + 6.85e13i)11-s + (4.20e13 + 4.04e14i)12-s + (−1.09e15 − 1.09e15i)13-s + (1.01e15 + 3.60e14i)14-s − 6.57e15·15-s + (3.70e15 + 1.76e16i)16-s + 5.24e16·17-s + ⋯
L(s)  = 1  + (0.334 − 0.942i)2-s + (−0.775 − 0.775i)3-s + (−0.776 − 0.630i)4-s + (0.562 − 0.562i)5-s + (−0.990 + 0.471i)6-s + 0.362i·7-s + (−0.853 + 0.521i)8-s + 0.203i·9-s + (−0.342 − 0.718i)10-s + (−0.598 + 0.598i)11-s + (0.113 + 1.09i)12-s + (−0.999 − 0.999i)13-s + (0.341 + 0.121i)14-s − 0.872·15-s + (0.205 + 0.978i)16-s + 1.28·17-s + ⋯

Functional equation

Λ(s)=(16s/2 ΓC(s) L(s)=((0.564+0.825i) Λ‾(28−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(28-s) \end{aligned}
Λ(s)=(16s/2 ΓC(s+27/2) L(s)=((0.564+0.825i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1616    =    242^{4}
Sign: 0.564+0.825i0.564 + 0.825i
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), 0.564+0.825i)(2,\ 16,\ (\ :27/2),\ 0.564 + 0.825i)

Particular Values

L(14)L(14) ≈\approx 1.2076151351.207615135
L(12)L(\frac12) ≈\approx 1.2076151351.207615135
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+(−3.87e3+1.09e4i)T 1 + (-3.87e3 + 1.09e4i)T
good3 1+(2.14e6+2.14e6i)T+7.62e12iT2 1 + (2.14e6 + 2.14e6i)T + 7.62e12iT^{2}
5 1+(−1.53e9+1.53e9i)T−7.45e18iT2 1 + (-1.53e9 + 1.53e9i)T - 7.45e18iT^{2}
7 1−9.29e10iT−6.57e22T2 1 - 9.29e10iT - 6.57e22T^{2}
11 1+(6.85e13−6.85e13i)T−1.31e28iT2 1 + (6.85e13 - 6.85e13i)T - 1.31e28iT^{2}
13 1+(1.09e15+1.09e15i)T+1.19e30iT2 1 + (1.09e15 + 1.09e15i)T + 1.19e30iT^{2}
17 1−5.24e16T+1.66e33T2 1 - 5.24e16T + 1.66e33T^{2}
19 1+(−1.10e17−1.10e17i)T+3.36e34iT2 1 + (-1.10e17 - 1.10e17i)T + 3.36e34iT^{2}
23 1−2.41e18iT−5.84e36T2 1 - 2.41e18iT - 5.84e36T^{2}
29 1+(−3.79e19−3.79e19i)T+3.05e39iT2 1 + (-3.79e19 - 3.79e19i)T + 3.05e39iT^{2}
31 1+2.62e19T+1.84e40T2 1 + 2.62e19T + 1.84e40T^{2}
37 1+(1.08e21−1.08e21i)T−2.19e42iT2 1 + (1.08e21 - 1.08e21i)T - 2.19e42iT^{2}
41 1+4.76e21iT−3.50e43T2 1 + 4.76e21iT - 3.50e43T^{2}
43 1+(−9.21e21+9.21e21i)T−1.26e44iT2 1 + (-9.21e21 + 9.21e21i)T - 1.26e44iT^{2}
47 1+1.68e21T+1.40e45T2 1 + 1.68e21T + 1.40e45T^{2}
53 1+(6.29e22−6.29e22i)T−3.59e46iT2 1 + (6.29e22 - 6.29e22i)T - 3.59e46iT^{2}
59 1+(7.91e23−7.91e23i)T−6.50e47iT2 1 + (7.91e23 - 7.91e23i)T - 6.50e47iT^{2}
61 1+(6.67e23+6.67e23i)T+1.59e48iT2 1 + (6.67e23 + 6.67e23i)T + 1.59e48iT^{2}
67 1+(3.91e24+3.91e24i)T+2.01e49iT2 1 + (3.91e24 + 3.91e24i)T + 2.01e49iT^{2}
71 1−3.66e24iT−9.63e49T2 1 - 3.66e24iT - 9.63e49T^{2}
73 1+2.35e24iT−2.04e50T2 1 + 2.35e24iT - 2.04e50T^{2}
79 1−3.15e25T+1.72e51T2 1 - 3.15e25T + 1.72e51T^{2}
83 1+(−9.31e25−9.31e25i)T+6.53e51iT2 1 + (-9.31e25 - 9.31e25i)T + 6.53e51iT^{2}
89 1+2.63e26iT−4.30e52T2 1 + 2.63e26iT - 4.30e52T^{2}
97 1+1.60e25T+4.39e53T2 1 + 1.60e25T + 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

−12.33269836989662281258504520160, −12.26540959432471022853037808192, −10.44308021476296877815869146133, −9.396073310656457900374302960697, −7.56914619556892848955171713876, −5.63835062414147678606913865351, −5.20896005015662794926260790204, −3.16350156881937103193129127171, −1.74532930600629333452624398630, −0.867411249936093623985480726274, 0.36984222389893286625630314997, 2.77336866124200143368850167166, 4.36444919591846181603385218167, 5.33266993203777442105300665117, 6.42094311887759022224465687105, 7.73499733802200691347321689624, 9.527753318223561471567586835045, 10.54683001958876509009885528770, 12.05717822230937319248976171182, 13.72411278437123224598515039144

Graph of the ZZ-function along the critical line