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Properties

Label 2-54-27.2-c2-0-2
Degree 22
Conductor 5454
Sign 0.750+0.660i0.750 + 0.660i
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 + (2.66 − 1.37i)3-s + (1.87 + 0.684i)4-s + (−1.49 − 1.77i)5-s + (−4.05 + 1.26i)6-s + (5.91 − 2.15i)7-s + (−2.44 − 1.41i)8-s + (5.20 − 7.33i)9-s + (1.64 + 2.84i)10-s + (1.00 − 1.20i)11-s + (5.95 − 0.764i)12-s + (1.33 + 7.56i)13-s + (−8.76 + 1.54i)14-s + (−6.43 − 2.68i)15-s + (3.06 + 2.57i)16-s + (−20.1 + 11.6i)17-s + ⋯
L(s)  = 1  + (−0.696 − 0.122i)2-s + (0.888 − 0.458i)3-s + (0.469 + 0.171i)4-s + (−0.298 − 0.355i)5-s + (−0.675 + 0.210i)6-s + (0.844 − 0.307i)7-s + (−0.306 − 0.176i)8-s + (0.578 − 0.815i)9-s + (0.164 + 0.284i)10-s + (0.0916 − 0.109i)11-s + (0.495 − 0.0637i)12-s + (0.102 + 0.581i)13-s + (−0.625 + 0.110i)14-s + (−0.428 − 0.179i)15-s + (0.191 + 0.160i)16-s + (−1.18 + 0.684i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.750+0.660i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((0.750+0.660i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.750+0.660i0.750 + 0.660i
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), 0.750+0.660i)(2,\ 54,\ (\ :1),\ 0.750 + 0.660i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.03780−0.391894i1.03780 - 0.391894i
L(12)L(\frac12) ≈\approx 1.03780−0.391894i1.03780 - 0.391894i
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+(−2.66+1.37i)T 1 + (-2.66 + 1.37i)T
good5 1+(1.49+1.77i)T+(−4.34+24.6i)T2 1 + (1.49 + 1.77i)T + (-4.34 + 24.6i)T^{2}
7 1+(−5.91+2.15i)T+(37.5−31.4i)T2 1 + (-5.91 + 2.15i)T + (37.5 - 31.4i)T^{2}
11 1+(−1.00+1.20i)T+(−21.0−119.i)T2 1 + (-1.00 + 1.20i)T + (-21.0 - 119. i)T^{2}
13 1+(−1.33−7.56i)T+(−158.+57.8i)T2 1 + (-1.33 - 7.56i)T + (-158. + 57.8i)T^{2}
17 1+(20.1−11.6i)T+(144.5−250.i)T2 1 + (20.1 - 11.6i)T + (144.5 - 250. i)T^{2}
19 1+(15.0−26.1i)T+(−180.5−312.i)T2 1 + (15.0 - 26.1i)T + (-180.5 - 312. i)T^{2}
23 1+(7.69−21.1i)T+(−405.−340.i)T2 1 + (7.69 - 21.1i)T + (-405. - 340. i)T^{2}
29 1+(−49.0−8.65i)T+(790.+287.i)T2 1 + (-49.0 - 8.65i)T + (790. + 287. i)T^{2}
31 1+(27.8+10.1i)T+(736.+617.i)T2 1 + (27.8 + 10.1i)T + (736. + 617. i)T^{2}
37 1+(−14.5−25.2i)T+(−684.5+1.18e3i)T2 1 + (-14.5 - 25.2i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(−17.1+3.02i)T+(1.57e3−574.i)T2 1 + (-17.1 + 3.02i)T + (1.57e3 - 574. i)T^{2}
43 1+(64.1+53.7i)T+(321.+1.82e3i)T2 1 + (64.1 + 53.7i)T + (321. + 1.82e3i)T^{2}
47 1+(−11.3−31.1i)T+(−1.69e3+1.41e3i)T2 1 + (-11.3 - 31.1i)T + (-1.69e3 + 1.41e3i)T^{2}
53 1+86.0iT−2.80e3T2 1 + 86.0iT - 2.80e3T^{2}
59 1+(−29.2−34.9i)T+(−604.+3.42e3i)T2 1 + (-29.2 - 34.9i)T + (-604. + 3.42e3i)T^{2}
61 1+(−79.4+28.9i)T+(2.85e3−2.39e3i)T2 1 + (-79.4 + 28.9i)T + (2.85e3 - 2.39e3i)T^{2}
67 1+(8.80+49.9i)T+(−4.21e3+1.53e3i)T2 1 + (8.80 + 49.9i)T + (-4.21e3 + 1.53e3i)T^{2}
71 1+(32.5−18.7i)T+(2.52e3−4.36e3i)T2 1 + (32.5 - 18.7i)T + (2.52e3 - 4.36e3i)T^{2}
73 1+(3.93−6.82i)T+(−2.66e3−4.61e3i)T2 1 + (3.93 - 6.82i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(12.5−71.1i)T+(−5.86e3−2.13e3i)T2 1 + (12.5 - 71.1i)T + (-5.86e3 - 2.13e3i)T^{2}
83 1+(25.4+4.47i)T+(6.47e3+2.35e3i)T2 1 + (25.4 + 4.47i)T + (6.47e3 + 2.35e3i)T^{2}
89 1+(23.4+13.5i)T+(3.96e3+6.85e3i)T2 1 + (23.4 + 13.5i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(27.0+22.6i)T+(1.63e3+9.26e3i)T2 1 + (27.0 + 22.6i)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

−14.94089866328298589184029510749, −13.95865412544738781321315496257, −12.66076444745489473456762280086, −11.52931635215837254157840032737, −10.13960870618966698778545021678, −8.647260881240783858309510930375, −8.086325948341846036816438161047, −6.63600865927231771640720021908, −4.07723058900772414390851245447, −1.78401267701721940044924095631, 2.55374762041958283778299267696, 4.68845264259736452791407064947, 6.95406418364898416851889688276, 8.254977594345423952659443509269, 9.064119107212924483221265574706, 10.50824199642014831231108996765, 11.39773471923444296542219603078, 13.15192355988045919847708771907, 14.52793772444276584532682942997, 15.24512312976223084000493504955

Graph of the ZZ-function along the critical line