↑

Properties

Label 2-54-27.11-c2-0-5
Degree 22
Conductor 5454
Sign 0.397+0.917i0.397 + 0.917i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.909 − 1.08i)2-s + (1.54 − 2.56i)3-s + (−0.347 − 1.96i)4-s + (−1.07 + 2.96i)5-s + (−1.37 − 4.01i)6-s + (−0.250 + 1.42i)7-s + (−2.44 − 1.41i)8-s + (−4.20 − 7.95i)9-s + (2.23 + 3.86i)10-s + (6.39 + 17.5i)11-s + (−5.59 − 2.15i)12-s + (10.5 − 8.88i)13-s + (1.31 + 1.56i)14-s + (5.94 + 7.36i)15-s + (−3.75 + 1.36i)16-s + (−16.3 + 9.44i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.516 − 0.856i)3-s + (−0.0868 − 0.492i)4-s + (−0.215 + 0.592i)5-s + (−0.229 − 0.668i)6-s + (−0.0357 + 0.202i)7-s + (−0.306 − 0.176i)8-s + (−0.466 − 0.884i)9-s + (0.223 + 0.386i)10-s + (0.581 + 1.59i)11-s + (−0.466 − 0.179i)12-s + (0.814 − 0.683i)13-s + (0.0936 + 0.111i)14-s + (0.396 + 0.490i)15-s + (−0.234 + 0.0855i)16-s + (−0.961 + 0.555i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.397+0.917i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((0.397+0.917i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.397+0.917i0.397 + 0.917i
Analytic conductor: 1.471391.47139
Root analytic conductor: 1.213011.21301
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ54(11,⋅)\chi_{54} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1), 0.397+0.917i)(2,\ 54,\ (\ :1),\ 0.397 + 0.917i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.29376−0.849561i1.29376 - 0.849561i
L(12)L(\frac12) ≈\approx 1.29376−0.849561i1.29376 - 0.849561i
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+(−0.909+1.08i)T 1 + (-0.909 + 1.08i)T
3 1+(−1.54+2.56i)T 1 + (-1.54 + 2.56i)T
good5 1+(1.07−2.96i)T+(−19.1−16.0i)T2 1 + (1.07 - 2.96i)T + (-19.1 - 16.0i)T^{2}
7 1+(0.250−1.42i)T+(−46.0−16.7i)T2 1 + (0.250 - 1.42i)T + (-46.0 - 16.7i)T^{2}
11 1+(−6.39−17.5i)T+(−92.6+77.7i)T2 1 + (-6.39 - 17.5i)T + (-92.6 + 77.7i)T^{2}
13 1+(−10.5+8.88i)T+(29.3−166.i)T2 1 + (-10.5 + 8.88i)T + (29.3 - 166. i)T^{2}
17 1+(16.3−9.44i)T+(144.5−250.i)T2 1 + (16.3 - 9.44i)T + (144.5 - 250. i)T^{2}
19 1+(1.14−1.98i)T+(−180.5−312.i)T2 1 + (1.14 - 1.98i)T + (-180.5 - 312. i)T^{2}
23 1+(26.5−4.68i)T+(497.−180.i)T2 1 + (26.5 - 4.68i)T + (497. - 180. i)T^{2}
29 1+(−17.2+20.5i)T+(−146.−828.i)T2 1 + (-17.2 + 20.5i)T + (-146. - 828. i)T^{2}
31 1+(4.55+25.8i)T+(−903.+328.i)T2 1 + (4.55 + 25.8i)T + (-903. + 328. i)T^{2}
37 1+(33.8+58.6i)T+(−684.5+1.18e3i)T2 1 + (33.8 + 58.6i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(13.0+15.5i)T+(−291.+1.65e3i)T2 1 + (13.0 + 15.5i)T + (-291. + 1.65e3i)T^{2}
43 1+(32.2−11.7i)T+(1.41e3−1.18e3i)T2 1 + (32.2 - 11.7i)T + (1.41e3 - 1.18e3i)T^{2}
47 1+(−46.4−8.19i)T+(2.07e3+755.i)T2 1 + (-46.4 - 8.19i)T + (2.07e3 + 755. i)T^{2}
53 1+49.0iT−2.80e3T2 1 + 49.0iT - 2.80e3T^{2}
59 1+(13.0−35.9i)T+(−2.66e3−2.23e3i)T2 1 + (13.0 - 35.9i)T + (-2.66e3 - 2.23e3i)T^{2}
61 1+(9.55−54.2i)T+(−3.49e3−1.27e3i)T2 1 + (9.55 - 54.2i)T + (-3.49e3 - 1.27e3i)T^{2}
67 1+(95.2−79.9i)T+(779.−4.42e3i)T2 1 + (95.2 - 79.9i)T + (779. - 4.42e3i)T^{2}
71 1+(−10.4+6.04i)T+(2.52e3−4.36e3i)T2 1 + (-10.4 + 6.04i)T + (2.52e3 - 4.36e3i)T^{2}
73 1+(−37.3+64.7i)T+(−2.66e3−4.61e3i)T2 1 + (-37.3 + 64.7i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(−74.8−62.7i)T+(1.08e3+6.14e3i)T2 1 + (-74.8 - 62.7i)T + (1.08e3 + 6.14e3i)T^{2}
83 1+(−81.1+96.6i)T+(−1.19e3−6.78e3i)T2 1 + (-81.1 + 96.6i)T + (-1.19e3 - 6.78e3i)T^{2}
89 1+(−9.23−5.33i)T+(3.96e3+6.85e3i)T2 1 + (-9.23 - 5.33i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(−145.+53.0i)T+(7.20e3−6.04e3i)T2 1 + (-145. + 53.0i)T + (7.20e3 - 6.04e3i)T^{2}
show more
show less
   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

−14.80493152420548777928989870828, −13.64901859511739893711104592282, −12.64936688495860256201038711321, −11.76204231759163151584847428070, −10.38337485194511327614197068157, −8.942280349696096109327819314063, −7.40549408252328593977674806805, −6.16748195500619952805040235798, −3.89381276017032085547010350794, −2.11540003380967289045356349542, 3.52243750840087464983666773797, 4.78639621197895058270507874891, 6.42521568881088734539860975734, 8.419559879556092681782845778218, 8.936488750296122830231855640717, 10.73322793784886599874390618335, 11.90882090216400100612005541930, 13.70456419201752749617229554513, 13.97180324320415481633378282619, 15.51237182317353032532532851263

Graph of the ZZ-function along the critical line