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Properties

Label 2-54-27.11-c8-0-14
Degree 22
Conductor 5454
Sign 0.607+0.794i0.607 + 0.794i
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 + (−36.0 + 72.5i)3-s + (−22.2 − 126. i)4-s + (0.862 − 2.37i)5-s + (366. + 839. i)6-s + (117. − 669. i)7-s + (−1.25e3 − 724. i)8-s + (−3.96e3 − 5.23e3i)9-s + (−14.2 − 24.7i)10-s + (3.73e3 + 1.02e4i)11-s + (9.94e3 + 2.93e3i)12-s + (−5.25e3 + 4.41e3i)13-s + (−4.94e3 − 5.88e3i)14-s + (140. + 148. i)15-s + (−1.53e4 + 5.60e3i)16-s + (9.61e4 − 5.54e4i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (−0.445 + 0.895i)3-s + (−0.0868 − 0.492i)4-s + (0.00138 − 0.00379i)5-s + (0.282 + 0.648i)6-s + (0.0491 − 0.278i)7-s + (−0.306 − 0.176i)8-s + (−0.603 − 0.797i)9-s + (−0.00142 − 0.00247i)10-s + (0.254 + 0.700i)11-s + (0.479 + 0.141i)12-s + (−0.184 + 0.154i)13-s + (−0.128 − 0.153i)14-s + (0.00278 + 0.00292i)15-s + (−0.234 + 0.0855i)16-s + (1.15 − 0.664i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.607+0.794i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+4) L(s)=((0.607+0.794i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.607+0.794i0.607 + 0.794i
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), 0.607+0.794i)(2,\ 54,\ (\ :4),\ 0.607 + 0.794i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 1.77231−0.876390i1.77231 - 0.876390i
L(12)L(\frac12) ≈\approx 1.77231−0.876390i1.77231 - 0.876390i
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+(36.0−72.5i)T 1 + (36.0 - 72.5i)T
good5 1+(−0.862+2.37i)T+(−2.99e5−2.51e5i)T2 1 + (-0.862 + 2.37i)T + (-2.99e5 - 2.51e5i)T^{2}
7 1+(−117.+669.i)T+(−5.41e6−1.97e6i)T2 1 + (-117. + 669. i)T + (-5.41e6 - 1.97e6i)T^{2}
11 1+(−3.73e3−1.02e4i)T+(−1.64e8+1.37e8i)T2 1 + (-3.73e3 - 1.02e4i)T + (-1.64e8 + 1.37e8i)T^{2}
13 1+(5.25e3−4.41e3i)T+(1.41e8−8.03e8i)T2 1 + (5.25e3 - 4.41e3i)T + (1.41e8 - 8.03e8i)T^{2}
17 1+(−9.61e4+5.54e4i)T+(3.48e9−6.04e9i)T2 1 + (-9.61e4 + 5.54e4i)T + (3.48e9 - 6.04e9i)T^{2}
19 1+(−1.20e5+2.09e5i)T+(−8.49e9−1.47e10i)T2 1 + (-1.20e5 + 2.09e5i)T + (-8.49e9 - 1.47e10i)T^{2}
23 1+(−9.71e4+1.71e4i)T+(7.35e10−2.67e10i)T2 1 + (-9.71e4 + 1.71e4i)T + (7.35e10 - 2.67e10i)T^{2}
29 1+(−3.29e5+3.92e5i)T+(−8.68e10−4.92e11i)T2 1 + (-3.29e5 + 3.92e5i)T + (-8.68e10 - 4.92e11i)T^{2}
31 1+(2.39e5+1.35e6i)T+(−8.01e11+2.91e11i)T2 1 + (2.39e5 + 1.35e6i)T + (-8.01e11 + 2.91e11i)T^{2}
37 1+(−6.30e5−1.09e6i)T+(−1.75e12+3.04e12i)T2 1 + (-6.30e5 - 1.09e6i)T + (-1.75e12 + 3.04e12i)T^{2}
41 1+(1.77e6+2.11e6i)T+(−1.38e12+7.86e12i)T2 1 + (1.77e6 + 2.11e6i)T + (-1.38e12 + 7.86e12i)T^{2}
43 1+(1.23e5−4.48e4i)T+(8.95e12−7.51e12i)T2 1 + (1.23e5 - 4.48e4i)T + (8.95e12 - 7.51e12i)T^{2}
47 1+(−1.64e6−2.89e5i)T+(2.23e13+8.14e12i)T2 1 + (-1.64e6 - 2.89e5i)T + (2.23e13 + 8.14e12i)T^{2}
53 1+6.15e6iT−6.22e13T2 1 + 6.15e6iT - 6.22e13T^{2}
59 1+(9.01e5−2.47e6i)T+(−1.12e14−9.43e13i)T2 1 + (9.01e5 - 2.47e6i)T + (-1.12e14 - 9.43e13i)T^{2}
61 1+(3.36e5−1.90e6i)T+(−1.80e14−6.55e13i)T2 1 + (3.36e5 - 1.90e6i)T + (-1.80e14 - 6.55e13i)T^{2}
67 1+(1.28e7−1.07e7i)T+(7.05e13−3.99e14i)T2 1 + (1.28e7 - 1.07e7i)T + (7.05e13 - 3.99e14i)T^{2}
71 1+(3.26e6−1.88e6i)T+(3.22e14−5.59e14i)T2 1 + (3.26e6 - 1.88e6i)T + (3.22e14 - 5.59e14i)T^{2}
73 1+(1.56e7−2.71e7i)T+(−4.03e14−6.98e14i)T2 1 + (1.56e7 - 2.71e7i)T + (-4.03e14 - 6.98e14i)T^{2}
79 1+(1.69e7+1.41e7i)T+(2.63e14+1.49e15i)T2 1 + (1.69e7 + 1.41e7i)T + (2.63e14 + 1.49e15i)T^{2}
83 1+(−1.89e7+2.26e7i)T+(−3.91e14−2.21e15i)T2 1 + (-1.89e7 + 2.26e7i)T + (-3.91e14 - 2.21e15i)T^{2}
89 1+(−1.93e7−1.11e7i)T+(1.96e15+3.40e15i)T2 1 + (-1.93e7 - 1.11e7i)T + (1.96e15 + 3.40e15i)T^{2}
97 1+(9.38e7−3.41e7i)T+(6.00e15−5.03e15i)T2 1 + (9.38e7 - 3.41e7i)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

−13.45949805058589343058100893956, −12.04334562207179142497449113551, −11.26985322081304057452690532294, −10.03053898156322007065172863435, −9.215674868689897486537272719855, −7.10028883742512344221767276223, −5.41322606678106994503994107948, −4.39977403609110820708820087103, −2.94618311867489712991654687113, −0.75385045735363171722518786766, 1.23099644412416907325072612883, 3.22680060865903575897799382847, 5.28655266714776007388777527677, 6.21436292847471236857966738997, 7.54713182224350653175589677557, 8.571653697953031019591146949699, 10.49659962295683198099281514593, 11.98558355249820981935759155540, 12.55744844687418902099979777333, 13.91932041362423918350778393296

Graph of the ZZ-function along the critical line