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Properties

Label 2-54-27.7-c1-0-2
Degree 22
Conductor 5454
Sign 0.835+0.549i0.835 + 0.549i
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  + (0.939 − 0.342i)2-s + (−1.11 − 1.32i)3-s + (0.766 − 0.642i)4-s + (0.439 + 2.49i)5-s + (−1.5 − 0.866i)6-s + (−1.79 − 1.50i)7-s + (0.500 − 0.866i)8-s + (−0.520 + 2.95i)9-s + (1.26 + 2.19i)10-s + (−0.745 + 4.22i)11-s + (−1.70 − 0.300i)12-s + (−0.713 − 0.259i)13-s + (−2.20 − 0.802i)14-s + (2.81 − 3.35i)15-s + (0.173 − 0.984i)16-s + (−2.46 − 4.26i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (−0.642 − 0.766i)3-s + (0.383 − 0.321i)4-s + (0.196 + 1.11i)5-s + (−0.612 − 0.353i)6-s + (−0.679 − 0.570i)7-s + (0.176 − 0.306i)8-s + (−0.173 + 0.984i)9-s + (0.400 + 0.693i)10-s + (−0.224 + 1.27i)11-s + (−0.492 − 0.0868i)12-s + (−0.197 − 0.0719i)13-s + (−0.589 − 0.214i)14-s + (0.727 − 0.867i)15-s + (0.0434 − 0.246i)16-s + (−0.596 − 1.03i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.835+0.549i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1/2) L(s)=((0.835+0.549i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.835+0.549i0.835 + 0.549i
Analytic conductor: 0.4311920.431192
Root analytic conductor: 0.6566520.656652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ54(7,⋅)\chi_{54} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1/2), 0.835+0.549i)(2,\ 54,\ (\ :1/2),\ 0.835 + 0.549i)

Particular Values

L(1)L(1) ≈\approx 0.929566−0.278293i0.929566 - 0.278293i
L(12)L(\frac12) ≈\approx 0.929566−0.278293i0.929566 - 0.278293i
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+(−0.939+0.342i)T 1 + (-0.939 + 0.342i)T
3 1+(1.11+1.32i)T 1 + (1.11 + 1.32i)T
good5 1+(−0.439−2.49i)T+(−4.69+1.71i)T2 1 + (-0.439 - 2.49i)T + (-4.69 + 1.71i)T^{2}
7 1+(1.79+1.50i)T+(1.21+6.89i)T2 1 + (1.79 + 1.50i)T + (1.21 + 6.89i)T^{2}
11 1+(0.745−4.22i)T+(−10.3−3.76i)T2 1 + (0.745 - 4.22i)T + (-10.3 - 3.76i)T^{2}
13 1+(0.713+0.259i)T+(9.95+8.35i)T2 1 + (0.713 + 0.259i)T + (9.95 + 8.35i)T^{2}
17 1+(2.46+4.26i)T+(−8.5+14.7i)T2 1 + (2.46 + 4.26i)T + (-8.5 + 14.7i)T^{2}
19 1+(−3.62+6.27i)T+(−9.5−16.4i)T2 1 + (-3.62 + 6.27i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.233−0.196i)T+(3.99−22.6i)T2 1 + (0.233 - 0.196i)T + (3.99 - 22.6i)T^{2}
29 1+(−2.91+1.06i)T+(22.2−18.6i)T2 1 + (-2.91 + 1.06i)T + (22.2 - 18.6i)T^{2}
31 1+(6.58−5.52i)T+(5.38−30.5i)T2 1 + (6.58 - 5.52i)T + (5.38 - 30.5i)T^{2}
37 1+(−3.78−6.55i)T+(−18.5+32.0i)T2 1 + (-3.78 - 6.55i)T + (-18.5 + 32.0i)T^{2}
41 1+(4.60+1.67i)T+(31.4+26.3i)T2 1 + (4.60 + 1.67i)T + (31.4 + 26.3i)T^{2}
43 1+(0.283−1.60i)T+(−40.4−14.7i)T2 1 + (0.283 - 1.60i)T + (-40.4 - 14.7i)T^{2}
47 1+(1.39+1.16i)T+(8.16+46.2i)T2 1 + (1.39 + 1.16i)T + (8.16 + 46.2i)T^{2}
53 1−0.573T+53T2 1 - 0.573T + 53T^{2}
59 1+(−0.950−5.39i)T+(−55.4+20.1i)T2 1 + (-0.950 - 5.39i)T + (-55.4 + 20.1i)T^{2}
61 1+(−8.46−7.10i)T+(10.5+60.0i)T2 1 + (-8.46 - 7.10i)T + (10.5 + 60.0i)T^{2}
67 1+(0.0393+0.0143i)T+(51.3+43.0i)T2 1 + (0.0393 + 0.0143i)T + (51.3 + 43.0i)T^{2}
71 1+(−2.10−3.64i)T+(−35.5+61.4i)T2 1 + (-2.10 - 3.64i)T + (-35.5 + 61.4i)T^{2}
73 1+(−5.54+9.60i)T+(−36.5−63.2i)T2 1 + (-5.54 + 9.60i)T + (-36.5 - 63.2i)T^{2}
79 1+(−6.92+2.52i)T+(60.5−50.7i)T2 1 + (-6.92 + 2.52i)T + (60.5 - 50.7i)T^{2}
83 1+(6.41−2.33i)T+(63.5−53.3i)T2 1 + (6.41 - 2.33i)T + (63.5 - 53.3i)T^{2}
89 1+(3.96−6.86i)T+(−44.5−77.0i)T2 1 + (3.96 - 6.86i)T + (-44.5 - 77.0i)T^{2}
97 1+(0.570−3.23i)T+(−91.1−33.1i)T2 1 + (0.570 - 3.23i)T + (-91.1 - 33.1i)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

−15.11869521607941332312603868635, −13.84702960009547673667086079729, −13.12558336336656766506595488042, −11.91714563610639375103515204358, −10.87797341539091563079830117386, −9.842787852491263120741460873091, −7.11390196016222587747188680258, −6.81778934042816611592621963328, −4.98056117730359741131778298575, −2.70182334989983768221891677530, 3.73362244416975614718113193614, 5.36002748445467669611286149486, 6.11644928847088868153130300080, 8.421337929443840754432665237503, 9.584600478947968357259233567496, 11.05447164511094643243591016677, 12.28217799209658423776321063989, 13.03477811383327586193709992224, 14.47011777915254830561903046478, 15.77621274435088217946463173599

Graph of the ZZ-function along the critical line