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Properties

Label 2-54-27.23-c2-0-3
Degree 22
Conductor 5454
Sign 0.997+0.0725i0.997 + 0.0725i
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 + (−0.391 − 2.97i)3-s + (−1.53 − 1.28i)4-s + (7.52 + 1.32i)5-s + (4.14 + 0.918i)6-s + (4.40 − 3.69i)7-s + (2.44 − 1.41i)8-s + (−8.69 + 2.33i)9-s + (−5.40 + 9.35i)10-s + (8.02 − 1.41i)11-s + (−3.22 + 5.06i)12-s + (−22.0 + 8.02i)13-s + (2.78 + 7.64i)14-s + (0.998 − 22.8i)15-s + (0.694 + 3.93i)16-s + (6.39 + 3.69i)17-s + ⋯
L(s)  = 1  + (−0.241 + 0.664i)2-s + (−0.130 − 0.991i)3-s + (−0.383 − 0.321i)4-s + (1.50 + 0.265i)5-s + (0.690 + 0.153i)6-s + (0.629 − 0.528i)7-s + (0.306 − 0.176i)8-s + (−0.965 + 0.258i)9-s + (−0.540 + 0.935i)10-s + (0.729 − 0.128i)11-s + (−0.268 + 0.421i)12-s + (−1.69 + 0.617i)13-s + (0.198 + 0.546i)14-s + (0.0665 − 1.52i)15-s + (0.0434 + 0.246i)16-s + (0.376 + 0.217i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.997+0.0725i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((0.997+0.0725i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.997+0.0725i0.997 + 0.0725i
Analytic conductor: 1.471391.47139
Root analytic conductor: 1.213011.21301
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ54(23,⋅)\chi_{54} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1), 0.997+0.0725i)(2,\ 54,\ (\ :1),\ 0.997 + 0.0725i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.15734−0.0420098i1.15734 - 0.0420098i
L(12)L(\frac12) ≈\approx 1.15734−0.0420098i1.15734 - 0.0420098i
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+(0.391+2.97i)T 1 + (0.391 + 2.97i)T
good5 1+(−7.52−1.32i)T+(23.4+8.55i)T2 1 + (-7.52 - 1.32i)T + (23.4 + 8.55i)T^{2}
7 1+(−4.40+3.69i)T+(8.50−48.2i)T2 1 + (-4.40 + 3.69i)T + (8.50 - 48.2i)T^{2}
11 1+(−8.02+1.41i)T+(113.−41.3i)T2 1 + (-8.02 + 1.41i)T + (113. - 41.3i)T^{2}
13 1+(22.0−8.02i)T+(129.−108.i)T2 1 + (22.0 - 8.02i)T + (129. - 108. i)T^{2}
17 1+(−6.39−3.69i)T+(144.5+250.i)T2 1 + (-6.39 - 3.69i)T + (144.5 + 250. i)T^{2}
19 1+(7.80+13.5i)T+(−180.5+312.i)T2 1 + (7.80 + 13.5i)T + (-180.5 + 312. i)T^{2}
23 1+(19.9−23.8i)T+(−91.8−520.i)T2 1 + (19.9 - 23.8i)T + (-91.8 - 520. i)T^{2}
29 1+(9.68−26.6i)T+(−644.−540.i)T2 1 + (9.68 - 26.6i)T + (-644. - 540. i)T^{2}
31 1+(−12.2−10.2i)T+(166.+946.i)T2 1 + (-12.2 - 10.2i)T + (166. + 946. i)T^{2}
37 1+(−5.99+10.3i)T+(−684.5−1.18e3i)T2 1 + (-5.99 + 10.3i)T + (-684.5 - 1.18e3i)T^{2}
41 1+(−2.82−7.76i)T+(−1.28e3+1.08e3i)T2 1 + (-2.82 - 7.76i)T + (-1.28e3 + 1.08e3i)T^{2}
43 1+(7.82+44.3i)T+(−1.73e3+632.i)T2 1 + (7.82 + 44.3i)T + (-1.73e3 + 632. i)T^{2}
47 1+(43.4+51.7i)T+(−383.+2.17e3i)T2 1 + (43.4 + 51.7i)T + (-383. + 2.17e3i)T^{2}
53 1−16.4iT−2.80e3T2 1 - 16.4iT - 2.80e3T^{2}
59 1+(−57.3−10.1i)T+(3.27e3+1.19e3i)T2 1 + (-57.3 - 10.1i)T + (3.27e3 + 1.19e3i)T^{2}
61 1+(−54.6+45.8i)T+(646.−3.66e3i)T2 1 + (-54.6 + 45.8i)T + (646. - 3.66e3i)T^{2}
67 1+(47.1−17.1i)T+(3.43e3−2.88e3i)T2 1 + (47.1 - 17.1i)T + (3.43e3 - 2.88e3i)T^{2}
71 1+(−35.4−20.4i)T+(2.52e3+4.36e3i)T2 1 + (-35.4 - 20.4i)T + (2.52e3 + 4.36e3i)T^{2}
73 1+(49.7+86.2i)T+(−2.66e3+4.61e3i)T2 1 + (49.7 + 86.2i)T + (-2.66e3 + 4.61e3i)T^{2}
79 1+(72.4+26.3i)T+(4.78e3+4.01e3i)T2 1 + (72.4 + 26.3i)T + (4.78e3 + 4.01e3i)T^{2}
83 1+(−36.0+99.1i)T+(−5.27e3−4.42e3i)T2 1 + (-36.0 + 99.1i)T + (-5.27e3 - 4.42e3i)T^{2}
89 1+(0.302−0.174i)T+(3.96e3−6.85e3i)T2 1 + (0.302 - 0.174i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(−11.1−62.9i)T+(−8.84e3+3.21e3i)T2 1 + (-11.1 - 62.9i)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

−14.58077349817188532013408969542, −14.17417273490219947001399850895, −13.18901748325731551696080311004, −11.76164261165693013207419396042, −10.21591979810096149385330540051, −9.051052491769334937187299131772, −7.45346337405598955660737900625, −6.52310837163220674091954218943, −5.22279972350027477935399161181, −1.85497383287076393216452859612, 2.36590486683348285496550295967, 4.66887554076336065137421856876, 5.86101738981120090715707502114, 8.372464282198834270060882717426, 9.753128063155606639688239273749, 10.00374708619459914866121433884, 11.59626516548291808063499681409, 12.63887665470485342755307541118, 14.23213424166924098921387922894, 14.80761738471473379537600217888

Graph of the ZZ-function along the critical line