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Properties

Label 2-2700-3.2-c2-0-34
Degree 22
Conductor 27002700
Sign ii
Analytic cond. 73.569673.5696
Root an. cond. 8.577278.57727
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  − 7.15·7-s + 5.06i·11-s + 3.12·13-s − 8.72i·17-s + 20.1·19-s − 14.7i·23-s + 39.7i·29-s − 39.3·31-s − 34.8·37-s − 13.2i·41-s + 66.6·43-s + 16.9i·47-s + 2.18·49-s − 4.62i·53-s + 25.7i·59-s + ⋯
L(s)  = 1  − 1.02·7-s + 0.460i·11-s + 0.240·13-s − 0.513i·17-s + 1.06·19-s − 0.640i·23-s + 1.37i·29-s − 1.27·31-s − 0.942·37-s − 0.323i·41-s + 1.54·43-s + 0.359i·47-s + 0.0445·49-s − 0.0873i·53-s + 0.436i·59-s + ⋯

Functional equation

Λ(s)=(2700s/2 ΓC(s) L(s)=(i Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2700s/2 ΓC(s+1) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27002700    =    22⋅33⋅522^{2} \cdot 3^{3} \cdot 5^{2}
Sign: ii
Analytic conductor: 73.569673.5696
Root analytic conductor: 8.577278.57727
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ2700(701,⋅)\chi_{2700} (701, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2700, ( :1), i)(2,\ 2700,\ (\ :1),\ i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.0650673541.065067354
L(12)L(\frac12) ≈\approx 1.0650673541.065067354
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
3 1 1
5 1 1
good7 1+7.15T+49T2 1 + 7.15T + 49T^{2}
11 1−5.06iT−121T2 1 - 5.06iT - 121T^{2}
13 1−3.12T+169T2 1 - 3.12T + 169T^{2}
17 1+8.72iT−289T2 1 + 8.72iT - 289T^{2}
19 1−20.1T+361T2 1 - 20.1T + 361T^{2}
23 1+14.7iT−529T2 1 + 14.7iT - 529T^{2}
29 1−39.7iT−841T2 1 - 39.7iT - 841T^{2}
31 1+39.3T+961T2 1 + 39.3T + 961T^{2}
37 1+34.8T+1.36e3T2 1 + 34.8T + 1.36e3T^{2}
41 1+13.2iT−1.68e3T2 1 + 13.2iT - 1.68e3T^{2}
43 1−66.6T+1.84e3T2 1 - 66.6T + 1.84e3T^{2}
47 1−16.9iT−2.20e3T2 1 - 16.9iT - 2.20e3T^{2}
53 1+4.62iT−2.80e3T2 1 + 4.62iT - 2.80e3T^{2}
59 1−25.7iT−3.48e3T2 1 - 25.7iT - 3.48e3T^{2}
61 1+12.1T+3.72e3T2 1 + 12.1T + 3.72e3T^{2}
67 1+106.T+4.48e3T2 1 + 106.T + 4.48e3T^{2}
71 1−101.iT−5.04e3T2 1 - 101. iT - 5.04e3T^{2}
73 1−23.2T+5.32e3T2 1 - 23.2T + 5.32e3T^{2}
79 1−66.5T+6.24e3T2 1 - 66.5T + 6.24e3T^{2}
83 1+144.iT−6.88e3T2 1 + 144. iT - 6.88e3T^{2}
89 1+154.iT−7.92e3T2 1 + 154. iT - 7.92e3T^{2}
97 1−175.T+9.40e3T2 1 - 175.T + 9.40e3T^{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

−8.708998240726924756604261309501, −7.37809717075041888596850145852, −7.17212408899163480649583958098, −6.15292004651090441203751021510, −5.43514934532452236371686198713, −4.52636604765844613678453560514, −3.49139664574763532968192392191, −2.85821292579746878124116397397, −1.60259326198301978677811521371, −0.29472335673806291694807542070, 0.931021552056745294780123316688, 2.23027330545164702421569079494, 3.36023176918669929384655724894, 3.80439333566493039030870099879, 5.06816852565246859853560033522, 5.89186355691402580983322896436, 6.43131279299327212312906004501, 7.41319655171403216542803211092, 7.980923059344348806781883849728, 9.095804991438078334135959343604

Graph of the ZZ-function along the critical line