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Properties

Label 2-54-27.5-c8-0-4
Degree 22
Conductor 5454
Sign −0.821−0.570i-0.821 - 0.570i
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 + (−71.9 + 37.1i)3-s + (−22.2 + 126. i)4-s + (242. + 667. i)5-s + (845. + 353. i)6-s + (702. + 3.98e3i)7-s + (1.25e3 − 724. i)8-s + (3.80e3 − 5.34e3i)9-s + (4.01e3 − 6.95e3i)10-s + (−675. + 1.85e3i)11-s + (−3.08e3 − 9.89e3i)12-s + (5.27e3 + 4.42e3i)13-s + (2.94e4 − 3.50e4i)14-s + (−4.22e4 − 3.89e4i)15-s + (−1.53e4 − 5.60e3i)16-s + (8.59e4 + 4.96e4i)17-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (−0.888 + 0.458i)3-s + (−0.0868 + 0.492i)4-s + (0.388 + 1.06i)5-s + (0.652 + 0.272i)6-s + (0.292 + 1.65i)7-s + (0.306 − 0.176i)8-s + (0.579 − 0.815i)9-s + (0.401 − 0.695i)10-s + (−0.0461 + 0.126i)11-s + (−0.148 − 0.477i)12-s + (0.184 + 0.154i)13-s + (0.765 − 0.912i)14-s + (−0.834 − 0.770i)15-s + (−0.234 − 0.0855i)16-s + (1.02 + 0.594i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((−0.821−0.570i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+4) L(s)=((−0.821−0.570i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: −0.821−0.570i-0.821 - 0.570i
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), −0.821−0.570i)(2,\ 54,\ (\ :4),\ -0.821 - 0.570i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.297921+0.951113i0.297921 + 0.951113i
L(12)L(\frac12) ≈\approx 0.297921+0.951113i0.297921 + 0.951113i
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+(71.9−37.1i)T 1 + (71.9 - 37.1i)T
good5 1+(−242.−667.i)T+(−2.99e5+2.51e5i)T2 1 + (-242. - 667. i)T + (-2.99e5 + 2.51e5i)T^{2}
7 1+(−702.−3.98e3i)T+(−5.41e6+1.97e6i)T2 1 + (-702. - 3.98e3i)T + (-5.41e6 + 1.97e6i)T^{2}
11 1+(675.−1.85e3i)T+(−1.64e8−1.37e8i)T2 1 + (675. - 1.85e3i)T + (-1.64e8 - 1.37e8i)T^{2}
13 1+(−5.27e3−4.42e3i)T+(1.41e8+8.03e8i)T2 1 + (-5.27e3 - 4.42e3i)T + (1.41e8 + 8.03e8i)T^{2}
17 1+(−8.59e4−4.96e4i)T+(3.48e9+6.04e9i)T2 1 + (-8.59e4 - 4.96e4i)T + (3.48e9 + 6.04e9i)T^{2}
19 1+(−7.98e4−1.38e5i)T+(−8.49e9+1.47e10i)T2 1 + (-7.98e4 - 1.38e5i)T + (-8.49e9 + 1.47e10i)T^{2}
23 1+(3.83e5+6.76e4i)T+(7.35e10+2.67e10i)T2 1 + (3.83e5 + 6.76e4i)T + (7.35e10 + 2.67e10i)T^{2}
29 1+(−5.84e5−6.96e5i)T+(−8.68e10+4.92e11i)T2 1 + (-5.84e5 - 6.96e5i)T + (-8.68e10 + 4.92e11i)T^{2}
31 1+(−7.09e4+4.02e5i)T+(−8.01e11−2.91e11i)T2 1 + (-7.09e4 + 4.02e5i)T + (-8.01e11 - 2.91e11i)T^{2}
37 1+(1.78e6−3.09e6i)T+(−1.75e12−3.04e12i)T2 1 + (1.78e6 - 3.09e6i)T + (-1.75e12 - 3.04e12i)T^{2}
41 1+(−1.55e6+1.85e6i)T+(−1.38e12−7.86e12i)T2 1 + (-1.55e6 + 1.85e6i)T + (-1.38e12 - 7.86e12i)T^{2}
43 1+(5.76e6+2.09e6i)T+(8.95e12+7.51e12i)T2 1 + (5.76e6 + 2.09e6i)T + (8.95e12 + 7.51e12i)T^{2}
47 1+(−3.95e6+6.98e5i)T+(2.23e13−8.14e12i)T2 1 + (-3.95e6 + 6.98e5i)T + (2.23e13 - 8.14e12i)T^{2}
53 1+9.23e6iT−6.22e13T2 1 + 9.23e6iT - 6.22e13T^{2}
59 1+(−2.34e6−6.44e6i)T+(−1.12e14+9.43e13i)T2 1 + (-2.34e6 - 6.44e6i)T + (-1.12e14 + 9.43e13i)T^{2}
61 1+(4.42e6+2.50e7i)T+(−1.80e14+6.55e13i)T2 1 + (4.42e6 + 2.50e7i)T + (-1.80e14 + 6.55e13i)T^{2}
67 1+(3.41e6+2.86e6i)T+(7.05e13+3.99e14i)T2 1 + (3.41e6 + 2.86e6i)T + (7.05e13 + 3.99e14i)T^{2}
71 1+(−1.78e7−1.03e7i)T+(3.22e14+5.59e14i)T2 1 + (-1.78e7 - 1.03e7i)T + (3.22e14 + 5.59e14i)T^{2}
73 1+(4.93e6+8.54e6i)T+(−4.03e14+6.98e14i)T2 1 + (4.93e6 + 8.54e6i)T + (-4.03e14 + 6.98e14i)T^{2}
79 1+(2.85e6−2.39e6i)T+(2.63e14−1.49e15i)T2 1 + (2.85e6 - 2.39e6i)T + (2.63e14 - 1.49e15i)T^{2}
83 1+(−2.82e6−3.37e6i)T+(−3.91e14+2.21e15i)T2 1 + (-2.82e6 - 3.37e6i)T + (-3.91e14 + 2.21e15i)T^{2}
89 1+(−5.18e7+2.99e7i)T+(1.96e15−3.40e15i)T2 1 + (-5.18e7 + 2.99e7i)T + (1.96e15 - 3.40e15i)T^{2}
97 1+(2.82e7+1.02e7i)T+(6.00e15+5.03e15i)T2 1 + (2.82e7 + 1.02e7i)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

−14.25995087164078488043399320204, −12.27873265765880220907455531988, −11.83677219371768758241477610190, −10.49634660511037665298233785108, −9.798862923455957304476295061814, −8.315352347971550299063801108107, −6.45628330455895063478363050743, −5.38687316894959664806331983044, −3.35374245251395298404125668504, −1.79372094690095291471864657033, 0.52223759005282823575074389053, 1.25819894759700699711819639935, 4.50352713414111150557129975963, 5.61448275406299435504806286656, 7.08294760078383474501326729969, 8.014092050433559868302566757968, 9.691455155439598835559730081909, 10.69840065538176900651555614794, 12.00587494182378387114147959203, 13.39459758177659909492720428588

Graph of the ZZ-function along the critical line