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Properties

Label 2-33-33.29-c3-0-4
Degree 22
Conductor 3333
Sign 0.629−0.777i0.629 - 0.777i
Analytic cond. 1.947061.94706
Root an. cond. 1.395371.39537
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.97 + 2.16i)2-s + (4.28 + 2.94i)3-s + (1.70 + 5.24i)4-s + (−10.1 − 13.9i)5-s + (6.38 + 18.0i)6-s + (−16.0 + 5.21i)7-s + (2.82 − 8.70i)8-s + (9.69 + 25.1i)9-s − 63.2i·10-s + (36.4 + 0.331i)11-s + (−8.11 + 27.4i)12-s + (−34.3 + 47.2i)13-s + (−59.0 − 19.1i)14-s + (−2.38 − 89.4i)15-s + (62.8 − 45.6i)16-s + (−1.44 + 1.05i)17-s + ⋯
L(s)  = 1  + (1.05 + 0.763i)2-s + (0.824 + 0.566i)3-s + (0.212 + 0.655i)4-s + (−0.905 − 1.24i)5-s + (0.434 + 1.22i)6-s + (−0.867 + 0.281i)7-s + (0.124 − 0.384i)8-s + (0.359 + 0.933i)9-s − 2.00i·10-s + (0.999 + 0.00908i)11-s + (−0.195 + 0.660i)12-s + (−0.732 + 1.00i)13-s + (−1.12 − 0.366i)14-s + (−0.0410 − 1.53i)15-s + (0.982 − 0.713i)16-s + (−0.0206 + 0.0149i)17-s + ⋯

Functional equation

Λ(s)=(33s/2 ΓC(s) L(s)=((0.629−0.777i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(33s/2 ΓC(s+3/2) L(s)=((0.629−0.777i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3333    =    3⋅113 \cdot 11
Sign: 0.629−0.777i0.629 - 0.777i
Analytic conductor: 1.947061.94706
Root analytic conductor: 1.395371.39537
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ33(29,⋅)\chi_{33} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 33, ( :3/2), 0.629−0.777i)(2,\ 33,\ (\ :3/2),\ 0.629 - 0.777i)

Particular Values

L(2)L(2) ≈\approx 1.89599+0.904804i1.89599 + 0.904804i
L(12)L(\frac12) ≈\approx 1.89599+0.904804i1.89599 + 0.904804i
L(52)L(\frac{5}{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)
bad3 1+(−4.28−2.94i)T 1 + (-4.28 - 2.94i)T
11 1+(−36.4−0.331i)T 1 + (-36.4 - 0.331i)T
good2 1+(−2.97−2.16i)T+(2.47+7.60i)T2 1 + (-2.97 - 2.16i)T + (2.47 + 7.60i)T^{2}
5 1+(10.1+13.9i)T+(−38.6+118.i)T2 1 + (10.1 + 13.9i)T + (-38.6 + 118. i)T^{2}
7 1+(16.0−5.21i)T+(277.−201.i)T2 1 + (16.0 - 5.21i)T + (277. - 201. i)T^{2}
13 1+(34.3−47.2i)T+(−678.−2.08e3i)T2 1 + (34.3 - 47.2i)T + (-678. - 2.08e3i)T^{2}
17 1+(1.44−1.05i)T+(1.51e3−4.67e3i)T2 1 + (1.44 - 1.05i)T + (1.51e3 - 4.67e3i)T^{2}
19 1+(22.6+7.35i)T+(5.54e3+4.03e3i)T2 1 + (22.6 + 7.35i)T + (5.54e3 + 4.03e3i)T^{2}
23 1+75.4iT−1.21e4T2 1 + 75.4iT - 1.21e4T^{2}
29 1+(−17.7−54.5i)T+(−1.97e4+1.43e4i)T2 1 + (-17.7 - 54.5i)T + (-1.97e4 + 1.43e4i)T^{2}
31 1+(−127.−92.9i)T+(9.20e3+2.83e4i)T2 1 + (-127. - 92.9i)T + (9.20e3 + 2.83e4i)T^{2}
37 1+(71.2+219.i)T+(−4.09e4+2.97e4i)T2 1 + (71.2 + 219. i)T + (-4.09e4 + 2.97e4i)T^{2}
41 1+(50.3−154.i)T+(−5.57e4−4.05e4i)T2 1 + (50.3 - 154. i)T + (-5.57e4 - 4.05e4i)T^{2}
43 1+128.iT−7.95e4T2 1 + 128. iT - 7.95e4T^{2}
47 1+(404.+131.i)T+(8.39e4+6.10e4i)T2 1 + (404. + 131. i)T + (8.39e4 + 6.10e4i)T^{2}
53 1+(−357.+492.i)T+(−4.60e4−1.41e5i)T2 1 + (-357. + 492. i)T + (-4.60e4 - 1.41e5i)T^{2}
59 1+(13.0−4.23i)T+(1.66e5−1.20e5i)T2 1 + (13.0 - 4.23i)T + (1.66e5 - 1.20e5i)T^{2}
61 1+(−455.−626.i)T+(−7.01e4+2.15e5i)T2 1 + (-455. - 626. i)T + (-7.01e4 + 2.15e5i)T^{2}
67 1−199.T+3.00e5T2 1 - 199.T + 3.00e5T^{2}
71 1+(304.+418.i)T+(−1.10e5+3.40e5i)T2 1 + (304. + 418. i)T + (-1.10e5 + 3.40e5i)T^{2}
73 1+(−930.+302.i)T+(3.14e5−2.28e5i)T2 1 + (-930. + 302. i)T + (3.14e5 - 2.28e5i)T^{2}
79 1+(486.−669.i)T+(−1.52e5−4.68e5i)T2 1 + (486. - 669. i)T + (-1.52e5 - 4.68e5i)T^{2}
83 1+(−443.+322.i)T+(1.76e5−5.43e5i)T2 1 + (-443. + 322. i)T + (1.76e5 - 5.43e5i)T^{2}
89 1−399.iT−7.04e5T2 1 - 399. iT - 7.04e5T^{2}
97 1+(717.+521.i)T+(2.82e5+8.68e5i)T2 1 + (717. + 521. i)T + (2.82e5 + 8.68e5i)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

−16.25895713414696321338377447398, −15.16056162069925470181916248203, −14.22705353975509790874966914224, −12.97943748555179547140958435321, −12.04654349153605282356401351730, −9.658669707271825406117471326026, −8.571565176201064749376012668417, −6.86919074240482039174424965310, −4.83902511651706330803983162769, −3.84892063279643382285790673381, 2.88869007775586541087724832811, 3.80520027502689347425150235771, 6.62021025151045956301235593227, 7.919045274774202063652683982059, 10.00619966012322696544410093775, 11.50571316827350810384735983053, 12.47353558695288736790390084791, 13.59011122868679582422998216743, 14.61522932598361066920579183424, 15.35955925284234418091993071736

Graph of the ZZ-function along the critical line