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Properties

Label 2-54-27.20-c2-0-5
Degree 22
Conductor 5454
Sign −0.883−0.468i-0.883 - 0.468i
Analytic cond. 1.471391.47139
Root an. cond. 1.213011.21301
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.483 − 1.32i)2-s + (−2.93 + 0.613i)3-s + (−1.53 + 1.28i)4-s + (−7.71 + 1.35i)5-s + (2.23 + 3.60i)6-s + (−0.690 − 0.579i)7-s + (2.44 + 1.41i)8-s + (8.24 − 3.60i)9-s + (5.53 + 9.59i)10-s + (−15.2 − 2.68i)11-s + (3.71 − 4.71i)12-s + (−0.854 − 0.310i)13-s + (−0.435 + 1.19i)14-s + (21.8 − 8.72i)15-s + (0.694 − 3.93i)16-s + (10.6 − 6.15i)17-s + ⋯
L(s)  = 1  + (−0.241 − 0.664i)2-s + (−0.978 + 0.204i)3-s + (−0.383 + 0.321i)4-s + (−1.54 + 0.271i)5-s + (0.372 + 0.600i)6-s + (−0.0986 − 0.0827i)7-s + (0.306 + 0.176i)8-s + (0.916 − 0.400i)9-s + (0.553 + 0.959i)10-s + (−1.38 − 0.244i)11-s + (0.309 − 0.392i)12-s + (−0.0657 − 0.0239i)13-s + (−0.0311 + 0.0855i)14-s + (1.45 − 0.581i)15-s + (0.0434 − 0.246i)16-s + (0.627 − 0.362i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((−0.883−0.468i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.468i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((−0.883−0.468i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: −0.883−0.468i-0.883 - 0.468i
Analytic conductor: 1.471391.47139
Root analytic conductor: 1.213011.21301
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ54(47,⋅)\chi_{54} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1), −0.883−0.468i)(2,\ 54,\ (\ :1),\ -0.883 - 0.468i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.00182995+0.00736161i0.00182995 + 0.00736161i
L(12)L(\frac12) ≈\approx 0.00182995+0.00736161i0.00182995 + 0.00736161i
L(2)L(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+(0.483+1.32i)T 1 + (0.483 + 1.32i)T
3 1+(2.93−0.613i)T 1 + (2.93 - 0.613i)T
good5 1+(7.71−1.35i)T+(23.4−8.55i)T2 1 + (7.71 - 1.35i)T + (23.4 - 8.55i)T^{2}
7 1+(0.690+0.579i)T+(8.50+48.2i)T2 1 + (0.690 + 0.579i)T + (8.50 + 48.2i)T^{2}
11 1+(15.2+2.68i)T+(113.+41.3i)T2 1 + (15.2 + 2.68i)T + (113. + 41.3i)T^{2}
13 1+(0.854+0.310i)T+(129.+108.i)T2 1 + (0.854 + 0.310i)T + (129. + 108. i)T^{2}
17 1+(−10.6+6.15i)T+(144.5−250.i)T2 1 + (-10.6 + 6.15i)T + (144.5 - 250. i)T^{2}
19 1+(5.40−9.36i)T+(−180.5−312.i)T2 1 + (5.40 - 9.36i)T + (-180.5 - 312. i)T^{2}
23 1+(21.0+25.1i)T+(−91.8+520.i)T2 1 + (21.0 + 25.1i)T + (-91.8 + 520. i)T^{2}
29 1+(−19.3−53.1i)T+(−644.+540.i)T2 1 + (-19.3 - 53.1i)T + (-644. + 540. i)T^{2}
31 1+(37.9−31.8i)T+(166.−946.i)T2 1 + (37.9 - 31.8i)T + (166. - 946. i)T^{2}
37 1+(17.4+30.2i)T+(−684.5+1.18e3i)T2 1 + (17.4 + 30.2i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(12.2−33.6i)T+(−1.28e3−1.08e3i)T2 1 + (12.2 - 33.6i)T + (-1.28e3 - 1.08e3i)T^{2}
43 1+(−7.20+40.8i)T+(−1.73e3−632.i)T2 1 + (-7.20 + 40.8i)T + (-1.73e3 - 632. i)T^{2}
47 1+(15.9−18.9i)T+(−383.−2.17e3i)T2 1 + (15.9 - 18.9i)T + (-383. - 2.17e3i)T^{2}
53 1+50.3iT−2.80e3T2 1 + 50.3iT - 2.80e3T^{2}
59 1+(−65.7+11.5i)T+(3.27e3−1.19e3i)T2 1 + (-65.7 + 11.5i)T + (3.27e3 - 1.19e3i)T^{2}
61 1+(18.7+15.7i)T+(646.+3.66e3i)T2 1 + (18.7 + 15.7i)T + (646. + 3.66e3i)T^{2}
67 1+(61.2+22.3i)T+(3.43e3+2.88e3i)T2 1 + (61.2 + 22.3i)T + (3.43e3 + 2.88e3i)T^{2}
71 1+(−24.4+14.1i)T+(2.52e3−4.36e3i)T2 1 + (-24.4 + 14.1i)T + (2.52e3 - 4.36e3i)T^{2}
73 1+(10.7−18.6i)T+(−2.66e3−4.61e3i)T2 1 + (10.7 - 18.6i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(27.2−9.90i)T+(4.78e3−4.01e3i)T2 1 + (27.2 - 9.90i)T + (4.78e3 - 4.01e3i)T^{2}
83 1+(0.0509+0.139i)T+(−5.27e3+4.42e3i)T2 1 + (0.0509 + 0.139i)T + (-5.27e3 + 4.42e3i)T^{2}
89 1+(79.3+45.7i)T+(3.96e3+6.85e3i)T2 1 + (79.3 + 45.7i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(17.4−98.9i)T+(−8.84e3−3.21e3i)T2 1 + (17.4 - 98.9i)T + (-8.84e3 - 3.21e3i)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

−14.59837766949787657656472169916, −12.71731725492399289918372556242, −12.07945703781928438147186662095, −10.91727637263826496530588210929, −10.30531593631983914363450302034, −8.345798908613965302197947976017, −7.14957062882141058282042341103, −5.07347808183806805051262766568, −3.55205614885124629113812173392, −0.008878731844194105501932087260, 4.33309145550125846062191673777, 5.71223418392830785362933180869, 7.40833568522916347895942264872, 8.062467583069944309398557322826, 10.00596258588665242886314820144, 11.27766965995082418894982355025, 12.27532279411946299297429240273, 13.34863016213685537678690633352, 15.30562485059346714811042749641, 15.71713081316890238044754973447

Graph of the ZZ-function along the critical line