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Properties

Label 2-54-27.22-c1-0-2
Degree 22
Conductor 5454
Sign 0.625+0.780i0.625 + 0.780i
Analytic cond. 0.4311920.431192
Root an. cond. 0.6566520.656652
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 − 0.642i)2-s + (−1.36 − 1.07i)3-s + (0.173 − 0.984i)4-s + (0.696 − 0.253i)5-s + (−1.73 + 0.0539i)6-s + (0.717 + 4.07i)7-s + (−0.500 − 0.866i)8-s + (0.703 + 2.91i)9-s + (0.370 − 0.641i)10-s + (−4.27 − 1.55i)11-s + (−1.29 + 1.15i)12-s + (0.662 + 0.556i)13-s + (3.16 + 2.65i)14-s + (−1.21 − 0.401i)15-s + (−0.939 − 0.342i)16-s + (2.17 − 3.77i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−0.785 − 0.618i)3-s + (0.0868 − 0.492i)4-s + (0.311 − 0.113i)5-s + (−0.706 + 0.0220i)6-s + (0.271 + 1.53i)7-s + (−0.176 − 0.306i)8-s + (0.234 + 0.972i)9-s + (0.117 − 0.202i)10-s + (−1.28 − 0.468i)11-s + (−0.372 + 0.333i)12-s + (0.183 + 0.154i)13-s + (0.846 + 0.709i)14-s + (−0.314 − 0.103i)15-s + (−0.234 − 0.0855i)16-s + (0.528 − 0.915i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.625+0.780i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1/2) L(s)=((0.625+0.780i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.625+0.780i0.625 + 0.780i
Analytic conductor: 0.4311920.431192
Root analytic conductor: 0.6566520.656652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ54(49,⋅)\chi_{54} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1/2), 0.625+0.780i)(2,\ 54,\ (\ :1/2),\ 0.625 + 0.780i)

Particular Values

L(1)L(1) ≈\approx 0.822422−0.395042i0.822422 - 0.395042i
L(12)L(\frac12) ≈\approx 0.822422−0.395042i0.822422 - 0.395042i
L(32)L(\frac{3}{2}) 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+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
3 1+(1.36+1.07i)T 1 + (1.36 + 1.07i)T
good5 1+(−0.696+0.253i)T+(3.83−3.21i)T2 1 + (-0.696 + 0.253i)T + (3.83 - 3.21i)T^{2}
7 1+(−0.717−4.07i)T+(−6.57+2.39i)T2 1 + (-0.717 - 4.07i)T + (-6.57 + 2.39i)T^{2}
11 1+(4.27+1.55i)T+(8.42+7.07i)T2 1 + (4.27 + 1.55i)T + (8.42 + 7.07i)T^{2}
13 1+(−0.662−0.556i)T+(2.25+12.8i)T2 1 + (-0.662 - 0.556i)T + (2.25 + 12.8i)T^{2}
17 1+(−2.17+3.77i)T+(−8.5−14.7i)T2 1 + (-2.17 + 3.77i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.777+1.34i)T+(−9.5+16.4i)T2 1 + (0.777 + 1.34i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.608−3.45i)T+(−21.6−7.86i)T2 1 + (0.608 - 3.45i)T + (-21.6 - 7.86i)T^{2}
29 1+(2.50−2.10i)T+(5.03−28.5i)T2 1 + (2.50 - 2.10i)T + (5.03 - 28.5i)T^{2}
31 1+(−1.85+10.5i)T+(−29.1−10.6i)T2 1 + (-1.85 + 10.5i)T + (-29.1 - 10.6i)T^{2}
37 1+(−0.880+1.52i)T+(−18.5−32.0i)T2 1 + (-0.880 + 1.52i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.97+1.65i)T+(7.11+40.3i)T2 1 + (1.97 + 1.65i)T + (7.11 + 40.3i)T^{2}
43 1+(−2.58−0.941i)T+(32.9+27.6i)T2 1 + (-2.58 - 0.941i)T + (32.9 + 27.6i)T^{2}
47 1+(−1.68−9.54i)T+(−44.1+16.0i)T2 1 + (-1.68 - 9.54i)T + (-44.1 + 16.0i)T^{2}
53 1−4.00T+53T2 1 - 4.00T + 53T^{2}
59 1+(1.34−0.489i)T+(45.1−37.9i)T2 1 + (1.34 - 0.489i)T + (45.1 - 37.9i)T^{2}
61 1+(0.751+4.26i)T+(−57.3+20.8i)T2 1 + (0.751 + 4.26i)T + (-57.3 + 20.8i)T^{2}
67 1+(−10.0−8.42i)T+(11.6+65.9i)T2 1 + (-10.0 - 8.42i)T + (11.6 + 65.9i)T^{2}
71 1+(−2.54+4.40i)T+(−35.5−61.4i)T2 1 + (-2.54 + 4.40i)T + (-35.5 - 61.4i)T^{2}
73 1+(−0.286−0.496i)T+(−36.5+63.2i)T2 1 + (-0.286 - 0.496i)T + (-36.5 + 63.2i)T^{2}
79 1+(−5.17+4.34i)T+(13.7−77.7i)T2 1 + (-5.17 + 4.34i)T + (13.7 - 77.7i)T^{2}
83 1+(−7.06+5.92i)T+(14.4−81.7i)T2 1 + (-7.06 + 5.92i)T + (14.4 - 81.7i)T^{2}
89 1+(−6.19−10.7i)T+(−44.5+77.0i)T2 1 + (-6.19 - 10.7i)T + (-44.5 + 77.0i)T^{2}
97 1+(5.40+1.96i)T+(74.3+62.3i)T2 1 + (5.40 + 1.96i)T + (74.3 + 62.3i)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

−15.26553877769395835152167961989, −13.71402322914293604695564151867, −12.88134730037421261247844071945, −11.83110003741093156904671035201, −11.06122960206069902803615442099, −9.481003735007022639138362787149, −7.80460859551780265837531810505, −5.88484242553135047079676747998, −5.24154657136006027338953748870, −2.39363991817582668294811212740, 3.94215172792586259298413553251, 5.22042601629672777934718492782, 6.65158643732470709345400657438, 8.002830441724452405031123696588, 10.20068395613446654696349418484, 10.65490836344145337633912221483, 12.27438631536301560065151817138, 13.38226561288601886899943686693, 14.49679005561854283864826627467, 15.60792211901933189114972986063

Graph of the ZZ-function along the critical line