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Properties

Label 2-54-27.2-c2-0-1
Degree 22
Conductor 5454
Sign 0.614−0.788i0.614 - 0.788i
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  + (1.39 + 0.245i)2-s + (−1.56 + 2.55i)3-s + (1.87 + 0.684i)4-s + (1.85 + 2.21i)5-s + (−2.81 + 3.17i)6-s + (2.17 − 0.791i)7-s + (2.44 + 1.41i)8-s + (−4.07 − 8.02i)9-s + (2.04 + 3.54i)10-s + (−0.401 + 0.478i)11-s + (−4.69 + 3.73i)12-s + (−4.06 − 23.0i)13-s + (3.22 − 0.568i)14-s + (−8.58 + 1.27i)15-s + (3.06 + 2.57i)16-s + (2.71 − 1.56i)17-s + ⋯
L(s)  = 1  + (0.696 + 0.122i)2-s + (−0.523 + 0.852i)3-s + (0.469 + 0.171i)4-s + (0.371 + 0.443i)5-s + (−0.468 + 0.529i)6-s + (0.310 − 0.113i)7-s + (0.306 + 0.176i)8-s + (−0.452 − 0.891i)9-s + (0.204 + 0.354i)10-s + (−0.0364 + 0.0434i)11-s + (−0.391 + 0.311i)12-s + (−0.312 − 1.77i)13-s + (0.230 − 0.0405i)14-s + (−0.572 + 0.0851i)15-s + (0.191 + 0.160i)16-s + (0.159 − 0.0922i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.614−0.788i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((0.614−0.788i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.614−0.788i0.614 - 0.788i
Analytic conductor: 1.471391.47139
Root analytic conductor: 1.213011.21301
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ54(29,⋅)\chi_{54} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1), 0.614−0.788i)(2,\ 54,\ (\ :1),\ 0.614 - 0.788i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.34354+0.656233i1.34354 + 0.656233i
L(12)L(\frac12) ≈\approx 1.34354+0.656233i1.34354 + 0.656233i
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+(−1.39−0.245i)T 1 + (-1.39 - 0.245i)T
3 1+(1.56−2.55i)T 1 + (1.56 - 2.55i)T
good5 1+(−1.85−2.21i)T+(−4.34+24.6i)T2 1 + (-1.85 - 2.21i)T + (-4.34 + 24.6i)T^{2}
7 1+(−2.17+0.791i)T+(37.5−31.4i)T2 1 + (-2.17 + 0.791i)T + (37.5 - 31.4i)T^{2}
11 1+(0.401−0.478i)T+(−21.0−119.i)T2 1 + (0.401 - 0.478i)T + (-21.0 - 119. i)T^{2}
13 1+(4.06+23.0i)T+(−158.+57.8i)T2 1 + (4.06 + 23.0i)T + (-158. + 57.8i)T^{2}
17 1+(−2.71+1.56i)T+(144.5−250.i)T2 1 + (-2.71 + 1.56i)T + (144.5 - 250. i)T^{2}
19 1+(−2.04+3.53i)T+(−180.5−312.i)T2 1 + (-2.04 + 3.53i)T + (-180.5 - 312. i)T^{2}
23 1+(15.5−42.6i)T+(−405.−340.i)T2 1 + (15.5 - 42.6i)T + (-405. - 340. i)T^{2}
29 1+(−19.6−3.46i)T+(790.+287.i)T2 1 + (-19.6 - 3.46i)T + (790. + 287. i)T^{2}
31 1+(42.6+15.5i)T+(736.+617.i)T2 1 + (42.6 + 15.5i)T + (736. + 617. i)T^{2}
37 1+(−18.7−32.5i)T+(−684.5+1.18e3i)T2 1 + (-18.7 - 32.5i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(45.6−8.05i)T+(1.57e3−574.i)T2 1 + (45.6 - 8.05i)T + (1.57e3 - 574. i)T^{2}
43 1+(−46.3−38.9i)T+(321.+1.82e3i)T2 1 + (-46.3 - 38.9i)T + (321. + 1.82e3i)T^{2}
47 1+(−1.06−2.91i)T+(−1.69e3+1.41e3i)T2 1 + (-1.06 - 2.91i)T + (-1.69e3 + 1.41e3i)T^{2}
53 1+79.9iT−2.80e3T2 1 + 79.9iT - 2.80e3T^{2}
59 1+(41.9+50.0i)T+(−604.+3.42e3i)T2 1 + (41.9 + 50.0i)T + (-604. + 3.42e3i)T^{2}
61 1+(33.1−12.0i)T+(2.85e3−2.39e3i)T2 1 + (33.1 - 12.0i)T + (2.85e3 - 2.39e3i)T^{2}
67 1+(4.92+27.9i)T+(−4.21e3+1.53e3i)T2 1 + (4.92 + 27.9i)T + (-4.21e3 + 1.53e3i)T^{2}
71 1+(70.7−40.8i)T+(2.52e3−4.36e3i)T2 1 + (70.7 - 40.8i)T + (2.52e3 - 4.36e3i)T^{2}
73 1+(12.6−21.8i)T+(−2.66e3−4.61e3i)T2 1 + (12.6 - 21.8i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(−11.5+65.5i)T+(−5.86e3−2.13e3i)T2 1 + (-11.5 + 65.5i)T + (-5.86e3 - 2.13e3i)T^{2}
83 1+(−49.1−8.66i)T+(6.47e3+2.35e3i)T2 1 + (-49.1 - 8.66i)T + (6.47e3 + 2.35e3i)T^{2}
89 1+(−111.−64.1i)T+(3.96e3+6.85e3i)T2 1 + (-111. - 64.1i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(−107.−90.5i)T+(1.63e3+9.26e3i)T2 1 + (-107. - 90.5i)T + (1.63e3 + 9.26e3i)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

−15.22497596551513550842612500670, −14.38061859726868332974800811368, −13.07370372014809379021288880818, −11.78342261320214771345683433646, −10.70270539208830089317235173992, −9.755448906587822172692975534851, −7.80014655059487420909978670640, −6.07257726493750000589360452713, −5.02302622160369566363924687004, −3.27936556529763431089611201038, 1.94395402201650362130461327538, 4.63955033684825289650563767454, 6.00955182321320734971830951379, 7.21892607507845779132002803433, 8.883365622216489015368833170959, 10.67231295341103942776613561237, 11.87964851835524931595408677656, 12.57729168392786659695421338675, 13.79393555398054120187863889657, 14.50746877639781297391740731437

Graph of the ZZ-function along the critical line