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Properties

Label 2-54-27.5-c8-0-11
Degree 22
Conductor 5454
Sign 0.981+0.192i0.981 + 0.192i
Analytic cond. 21.998421.9984
Root an. cond. 4.690244.69024
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.27 − 8.66i)2-s + (7.33 + 80.6i)3-s + (−22.2 + 126. i)4-s + (58.6 + 161. i)5-s + (645. − 650. i)6-s + (−371. − 2.10e3i)7-s + (1.25e3 − 724. i)8-s + (−6.45e3 + 1.18e3i)9-s + (969. − 1.67e3i)10-s + (6.80e3 − 1.87e4i)11-s + (−1.03e4 − 867. i)12-s + (−1.07e3 − 898. i)13-s + (−1.55e4 + 1.85e4i)14-s + (−1.25e4 + 5.91e3i)15-s + (−1.53e4 − 5.60e3i)16-s + (5.04e4 + 2.91e4i)17-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (0.0905 + 0.995i)3-s + (−0.0868 + 0.492i)4-s + (0.0938 + 0.257i)5-s + (0.498 − 0.501i)6-s + (−0.154 − 0.877i)7-s + (0.306 − 0.176i)8-s + (−0.983 + 0.180i)9-s + (0.0969 − 0.167i)10-s + (0.465 − 1.27i)11-s + (−0.498 − 0.0418i)12-s + (−0.0374 − 0.0314i)13-s + (−0.405 + 0.482i)14-s + (−0.248 + 0.116i)15-s + (−0.234 − 0.0855i)16-s + (0.604 + 0.348i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.981+0.192i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+4) L(s)=((0.981+0.192i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.981+0.192i0.981 + 0.192i
Analytic conductor: 21.998421.9984
Root analytic conductor: 4.690244.69024
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ54(5,⋅)\chi_{54} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :4), 0.981+0.192i)(2,\ 54,\ (\ :4),\ 0.981 + 0.192i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 1.49499−0.144914i1.49499 - 0.144914i
L(12)L(\frac12) ≈\approx 1.49499−0.144914i1.49499 - 0.144914i
L(5)L(5) 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+(7.27+8.66i)T 1 + (7.27 + 8.66i)T
3 1+(−7.33−80.6i)T 1 + (-7.33 - 80.6i)T
good5 1+(−58.6−161.i)T+(−2.99e5+2.51e5i)T2 1 + (-58.6 - 161. i)T + (-2.99e5 + 2.51e5i)T^{2}
7 1+(371.+2.10e3i)T+(−5.41e6+1.97e6i)T2 1 + (371. + 2.10e3i)T + (-5.41e6 + 1.97e6i)T^{2}
11 1+(−6.80e3+1.87e4i)T+(−1.64e8−1.37e8i)T2 1 + (-6.80e3 + 1.87e4i)T + (-1.64e8 - 1.37e8i)T^{2}
13 1+(1.07e3+898.i)T+(1.41e8+8.03e8i)T2 1 + (1.07e3 + 898. i)T + (1.41e8 + 8.03e8i)T^{2}
17 1+(−5.04e4−2.91e4i)T+(3.48e9+6.04e9i)T2 1 + (-5.04e4 - 2.91e4i)T + (3.48e9 + 6.04e9i)T^{2}
19 1+(−8.93e4−1.54e5i)T+(−8.49e9+1.47e10i)T2 1 + (-8.93e4 - 1.54e5i)T + (-8.49e9 + 1.47e10i)T^{2}
23 1+(1.28e5+2.26e4i)T+(7.35e10+2.67e10i)T2 1 + (1.28e5 + 2.26e4i)T + (7.35e10 + 2.67e10i)T^{2}
29 1+(−7.69e5−9.17e5i)T+(−8.68e10+4.92e11i)T2 1 + (-7.69e5 - 9.17e5i)T + (-8.68e10 + 4.92e11i)T^{2}
31 1+(−1.38e5+7.84e5i)T+(−8.01e11−2.91e11i)T2 1 + (-1.38e5 + 7.84e5i)T + (-8.01e11 - 2.91e11i)T^{2}
37 1+(−1.31e6+2.27e6i)T+(−1.75e12−3.04e12i)T2 1 + (-1.31e6 + 2.27e6i)T + (-1.75e12 - 3.04e12i)T^{2}
41 1+(2.32e5−2.77e5i)T+(−1.38e12−7.86e12i)T2 1 + (2.32e5 - 2.77e5i)T + (-1.38e12 - 7.86e12i)T^{2}
43 1+(−4.97e6−1.81e6i)T+(8.95e12+7.51e12i)T2 1 + (-4.97e6 - 1.81e6i)T + (8.95e12 + 7.51e12i)T^{2}
47 1+(−1.90e5+3.35e4i)T+(2.23e13−8.14e12i)T2 1 + (-1.90e5 + 3.35e4i)T + (2.23e13 - 8.14e12i)T^{2}
53 1−8.84e6iT−6.22e13T2 1 - 8.84e6iT - 6.22e13T^{2}
59 1+(−3.69e5−1.01e6i)T+(−1.12e14+9.43e13i)T2 1 + (-3.69e5 - 1.01e6i)T + (-1.12e14 + 9.43e13i)T^{2}
61 1+(−1.22e6−6.96e6i)T+(−1.80e14+6.55e13i)T2 1 + (-1.22e6 - 6.96e6i)T + (-1.80e14 + 6.55e13i)T^{2}
67 1+(−5.96e5−5.00e5i)T+(7.05e13+3.99e14i)T2 1 + (-5.96e5 - 5.00e5i)T + (7.05e13 + 3.99e14i)T^{2}
71 1+(2.19e7+1.26e7i)T+(3.22e14+5.59e14i)T2 1 + (2.19e7 + 1.26e7i)T + (3.22e14 + 5.59e14i)T^{2}
73 1+(1.40e7+2.43e7i)T+(−4.03e14+6.98e14i)T2 1 + (1.40e7 + 2.43e7i)T + (-4.03e14 + 6.98e14i)T^{2}
79 1+(−5.12e7+4.29e7i)T+(2.63e14−1.49e15i)T2 1 + (-5.12e7 + 4.29e7i)T + (2.63e14 - 1.49e15i)T^{2}
83 1+(−3.12e7−3.71e7i)T+(−3.91e14+2.21e15i)T2 1 + (-3.12e7 - 3.71e7i)T + (-3.91e14 + 2.21e15i)T^{2}
89 1+(−2.76e7+1.59e7i)T+(1.96e15−3.40e15i)T2 1 + (-2.76e7 + 1.59e7i)T + (1.96e15 - 3.40e15i)T^{2}
97 1+(1.10e7+4.02e6i)T+(6.00e15+5.03e15i)T2 1 + (1.10e7 + 4.02e6i)T + (6.00e15 + 5.03e15i)T^{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

−13.83735385121327424846906931651, −12.15708807259068298195141822143, −10.87493779398124480459047312019, −10.26951747329798352242367820613, −9.074952684679517980143291689658, −7.85300577542194374953030839825, −5.96608147630969281001313196392, −4.08883339474507246092321251623, −3.07669967887439319524130291682, −0.831071189573786450087928275666, 0.997594772795874764688210137680, 2.49472573222536393438904515181, 5.08607813982018902978395478331, 6.45931576488427194385826825772, 7.49431987106399448511695283294, 8.768854702782200426771553747121, 9.731454597532871776431529518832, 11.65571880064961694204317398388, 12.49228673332682350122385965156, 13.72957255512336127780697006354

Graph of the ZZ-function along the critical line