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Properties

Label 2-54-27.11-c8-0-7
Degree 22
Conductor 5454
Sign −0.0683−0.997i-0.0683 - 0.997i
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 + (30.8 + 74.8i)3-s + (−22.2 − 126. i)4-s + (187. − 513. i)5-s + (873. + 276. i)6-s + (−775. + 4.39e3i)7-s + (−1.25e3 − 724. i)8-s + (−4.65e3 + 4.62e3i)9-s + (−3.09e3 − 5.35e3i)10-s + (−1.40e3 − 3.85e3i)11-s + (8.75e3 − 5.55e3i)12-s + (−1.48e4 + 1.24e4i)13-s + (3.24e4 + 3.87e4i)14-s + (4.42e4 − 1.86e3i)15-s + (−1.53e4 + 5.60e3i)16-s + (8.97e4 − 5.18e4i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.381 + 0.924i)3-s + (−0.0868 − 0.492i)4-s + (0.299 − 0.822i)5-s + (0.674 + 0.213i)6-s + (−0.322 + 1.83i)7-s + (−0.306 − 0.176i)8-s + (−0.709 + 0.705i)9-s + (−0.309 − 0.535i)10-s + (−0.0958 − 0.263i)11-s + (0.422 − 0.268i)12-s + (−0.518 + 0.435i)13-s + (0.845 + 1.00i)14-s + (0.874 − 0.0369i)15-s + (−0.234 + 0.0855i)16-s + (1.07 − 0.620i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((−0.0683−0.997i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0683 - 0.997i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+4) L(s)=((−0.0683−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0683 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: −0.0683−0.997i-0.0683 - 0.997i
Analytic conductor: 21.998421.9984
Root analytic conductor: 4.690244.69024
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ54(11,⋅)\chi_{54} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :4), −0.0683−0.997i)(2,\ 54,\ (\ :4),\ -0.0683 - 0.997i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 1.35719+1.45341i1.35719 + 1.45341i
L(12)L(\frac12) ≈\approx 1.35719+1.45341i1.35719 + 1.45341i
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+(−30.8−74.8i)T 1 + (-30.8 - 74.8i)T
good5 1+(−187.+513.i)T+(−2.99e5−2.51e5i)T2 1 + (-187. + 513. i)T + (-2.99e5 - 2.51e5i)T^{2}
7 1+(775.−4.39e3i)T+(−5.41e6−1.97e6i)T2 1 + (775. - 4.39e3i)T + (-5.41e6 - 1.97e6i)T^{2}
11 1+(1.40e3+3.85e3i)T+(−1.64e8+1.37e8i)T2 1 + (1.40e3 + 3.85e3i)T + (-1.64e8 + 1.37e8i)T^{2}
13 1+(1.48e4−1.24e4i)T+(1.41e8−8.03e8i)T2 1 + (1.48e4 - 1.24e4i)T + (1.41e8 - 8.03e8i)T^{2}
17 1+(−8.97e4+5.18e4i)T+(3.48e9−6.04e9i)T2 1 + (-8.97e4 + 5.18e4i)T + (3.48e9 - 6.04e9i)T^{2}
19 1+(1.17e5−2.03e5i)T+(−8.49e9−1.47e10i)T2 1 + (1.17e5 - 2.03e5i)T + (-8.49e9 - 1.47e10i)T^{2}
23 1+(1.39e5−2.45e4i)T+(7.35e10−2.67e10i)T2 1 + (1.39e5 - 2.45e4i)T + (7.35e10 - 2.67e10i)T^{2}
29 1+(3.66e5−4.36e5i)T+(−8.68e10−4.92e11i)T2 1 + (3.66e5 - 4.36e5i)T + (-8.68e10 - 4.92e11i)T^{2}
31 1+(−2.71e5−1.54e6i)T+(−8.01e11+2.91e11i)T2 1 + (-2.71e5 - 1.54e6i)T + (-8.01e11 + 2.91e11i)T^{2}
37 1+(−1.12e6−1.95e6i)T+(−1.75e12+3.04e12i)T2 1 + (-1.12e6 - 1.95e6i)T + (-1.75e12 + 3.04e12i)T^{2}
41 1+(3.31e6+3.95e6i)T+(−1.38e12+7.86e12i)T2 1 + (3.31e6 + 3.95e6i)T + (-1.38e12 + 7.86e12i)T^{2}
43 1+(−2.41e6+8.80e5i)T+(8.95e12−7.51e12i)T2 1 + (-2.41e6 + 8.80e5i)T + (8.95e12 - 7.51e12i)T^{2}
47 1+(−6.74e6−1.18e6i)T+(2.23e13+8.14e12i)T2 1 + (-6.74e6 - 1.18e6i)T + (2.23e13 + 8.14e12i)T^{2}
53 1+9.71e6iT−6.22e13T2 1 + 9.71e6iT - 6.22e13T^{2}
59 1+(2.73e6−7.50e6i)T+(−1.12e14−9.43e13i)T2 1 + (2.73e6 - 7.50e6i)T + (-1.12e14 - 9.43e13i)T^{2}
61 1+(3.74e5−2.12e6i)T+(−1.80e14−6.55e13i)T2 1 + (3.74e5 - 2.12e6i)T + (-1.80e14 - 6.55e13i)T^{2}
67 1+(−8.75e6+7.34e6i)T+(7.05e13−3.99e14i)T2 1 + (-8.75e6 + 7.34e6i)T + (7.05e13 - 3.99e14i)T^{2}
71 1+(−5.23e6+3.02e6i)T+(3.22e14−5.59e14i)T2 1 + (-5.23e6 + 3.02e6i)T + (3.22e14 - 5.59e14i)T^{2}
73 1+(−1.90e7+3.29e7i)T+(−4.03e14−6.98e14i)T2 1 + (-1.90e7 + 3.29e7i)T + (-4.03e14 - 6.98e14i)T^{2}
79 1+(1.72e7+1.44e7i)T+(2.63e14+1.49e15i)T2 1 + (1.72e7 + 1.44e7i)T + (2.63e14 + 1.49e15i)T^{2}
83 1+(−3.02e6+3.60e6i)T+(−3.91e14−2.21e15i)T2 1 + (-3.02e6 + 3.60e6i)T + (-3.91e14 - 2.21e15i)T^{2}
89 1+(7.31e6+4.22e6i)T+(1.96e15+3.40e15i)T2 1 + (7.31e6 + 4.22e6i)T + (1.96e15 + 3.40e15i)T^{2}
97 1+(−5.35e7+1.94e7i)T+(6.00e15−5.03e15i)T2 1 + (-5.35e7 + 1.94e7i)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

−14.05010576435618555246843128694, −12.51436964954659235730093465531, −11.94550450571719927897468677815, −10.27843474795272566869730806371, −9.267011426318175592035523236885, −8.491223124125208679198069553123, −5.76732637173604404000212615095, −5.01457832461848556349516364668, −3.31759894177991294763997888793, −1.98721456761584310916131246306, 0.54344446707692782547095444268, 2.61810637353338520584852118534, 4.05992248348343776200902257012, 6.18702294908066301225746867598, 7.14572358709317984087919009652, 7.87513248773061581535021754535, 9.826444610027161771602654195983, 11.05030948892874545213719882721, 12.71202210721749701205718725392, 13.46755186028223101631771306735

Graph of the ZZ-function along the critical line