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Properties

Label 2-54-9.2-c8-0-7
Degree 22
Conductor 5454
Sign 0.754+0.656i0.754 + 0.656i
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  + (9.79 + 5.65i)2-s + (63.9 + 110. i)4-s + (719. − 415. i)5-s + (1.34e3 − 2.32e3i)7-s + 1.44e3i·8-s + 9.39e3·10-s + (−2.13e4 − 1.23e4i)11-s + (−4.84e3 − 8.39e3i)13-s + (2.63e4 − 1.51e4i)14-s + (−8.19e3 + 1.41e4i)16-s − 4.26e4i·17-s + 1.01e5·19-s + (9.20e4 + 5.31e4i)20-s + (−1.39e5 − 2.41e5i)22-s + (3.96e5 − 2.29e5i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (1.15 − 0.664i)5-s + (0.559 − 0.969i)7-s + 0.353i·8-s + 0.939·10-s + (−1.45 − 0.842i)11-s + (−0.169 − 0.294i)13-s + (0.685 − 0.395i)14-s + (−0.125 + 0.216i)16-s − 0.510i·17-s + 0.776·19-s + (0.575 + 0.332i)20-s + (−0.595 − 1.03i)22-s + (1.41 − 0.819i)23-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.754+0.656i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+4) L(s)=((0.754+0.656i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.754+0.656i0.754 + 0.656i
Analytic conductor: 21.998421.9984
Root analytic conductor: 4.690244.69024
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ54(35,⋅)\chi_{54} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :4), 0.754+0.656i)(2,\ 54,\ (\ :4),\ 0.754 + 0.656i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 3.01642−1.12870i3.01642 - 1.12870i
L(12)L(\frac12) ≈\approx 3.01642−1.12870i3.01642 - 1.12870i
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+(−9.79−5.65i)T 1 + (-9.79 - 5.65i)T
3 1 1
good5 1+(−719.+415.i)T+(1.95e5−3.38e5i)T2 1 + (-719. + 415. i)T + (1.95e5 - 3.38e5i)T^{2}
7 1+(−1.34e3+2.32e3i)T+(−2.88e6−4.99e6i)T2 1 + (-1.34e3 + 2.32e3i)T + (-2.88e6 - 4.99e6i)T^{2}
11 1+(2.13e4+1.23e4i)T+(1.07e8+1.85e8i)T2 1 + (2.13e4 + 1.23e4i)T + (1.07e8 + 1.85e8i)T^{2}
13 1+(4.84e3+8.39e3i)T+(−4.07e8+7.06e8i)T2 1 + (4.84e3 + 8.39e3i)T + (-4.07e8 + 7.06e8i)T^{2}
17 1+4.26e4iT−6.97e9T2 1 + 4.26e4iT - 6.97e9T^{2}
19 1−1.01e5T+1.69e10T2 1 - 1.01e5T + 1.69e10T^{2}
23 1+(−3.96e5+2.29e5i)T+(3.91e10−6.78e10i)T2 1 + (-3.96e5 + 2.29e5i)T + (3.91e10 - 6.78e10i)T^{2}
29 1+(5.77e5+3.33e5i)T+(2.50e11+4.33e11i)T2 1 + (5.77e5 + 3.33e5i)T + (2.50e11 + 4.33e11i)T^{2}
31 1+(−7.09e5−1.22e6i)T+(−4.26e11+7.38e11i)T2 1 + (-7.09e5 - 1.22e6i)T + (-4.26e11 + 7.38e11i)T^{2}
37 1−1.62e6T+3.51e12T2 1 - 1.62e6T + 3.51e12T^{2}
41 1+(1.99e6−1.15e6i)T+(3.99e12−6.91e12i)T2 1 + (1.99e6 - 1.15e6i)T + (3.99e12 - 6.91e12i)T^{2}
43 1+(−8.12e5+1.40e6i)T+(−5.84e12−1.01e13i)T2 1 + (-8.12e5 + 1.40e6i)T + (-5.84e12 - 1.01e13i)T^{2}
47 1+(−3.38e6−1.95e6i)T+(1.19e13+2.06e13i)T2 1 + (-3.38e6 - 1.95e6i)T + (1.19e13 + 2.06e13i)T^{2}
53 1+6.96e6iT−6.22e13T2 1 + 6.96e6iT - 6.22e13T^{2}
59 1+(5.94e6−3.43e6i)T+(7.34e13−1.27e14i)T2 1 + (5.94e6 - 3.43e6i)T + (7.34e13 - 1.27e14i)T^{2}
61 1+(−1.26e6+2.19e6i)T+(−9.58e13−1.66e14i)T2 1 + (-1.26e6 + 2.19e6i)T + (-9.58e13 - 1.66e14i)T^{2}
67 1+(−1.58e7−2.74e7i)T+(−2.03e14+3.51e14i)T2 1 + (-1.58e7 - 2.74e7i)T + (-2.03e14 + 3.51e14i)T^{2}
71 1−1.75e7iT−6.45e14T2 1 - 1.75e7iT - 6.45e14T^{2}
73 1+2.86e7T+8.06e14T2 1 + 2.86e7T + 8.06e14T^{2}
79 1+(3.40e7−5.89e7i)T+(−7.58e14−1.31e15i)T2 1 + (3.40e7 - 5.89e7i)T + (-7.58e14 - 1.31e15i)T^{2}
83 1+(4.73e7+2.73e7i)T+(1.12e15+1.95e15i)T2 1 + (4.73e7 + 2.73e7i)T + (1.12e15 + 1.95e15i)T^{2}
89 1−7.04e6iT−3.93e15T2 1 - 7.04e6iT - 3.93e15T^{2}
97 1+(−5.14e7+8.90e7i)T+(−3.91e15−6.78e15i)T2 1 + (-5.14e7 + 8.90e7i)T + (-3.91e15 - 6.78e15i)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

−13.49679499652041941681576420332, −12.88479840138768157177349098760, −11.17474370657635960331854789148, −10.07875565105555132091868788644, −8.524858587154768753514844764956, −7.27072837138666488057228532251, −5.61831324591153852023572183178, −4.81143947532103271192421168828, −2.80941538971925220039871786973, −0.946461126396598787267795307644, 1.87670691091447628561302540890, 2.77931110449134155819151645680, 5.00546735566145990168775543498, 5.88973085127238475395609595581, 7.48013760926215557061714983954, 9.340221209621029243296073016408, 10.36258177087160878015181047433, 11.48164281985268626661427857718, 12.79890976749064487084459132320, 13.65297970088401341226153701786

Graph of the ZZ-function along the critical line