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Properties

Label 2-1872-1.1-c3-0-43
Degree 22
Conductor 18721872
Sign −1-1
Analytic cond. 110.451110.451
Root an. cond. 10.509510.5095
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 19.3·5-s − 4.84·7-s − 61.0·11-s + 13·13-s + 41.7·17-s + 107.·19-s + 28.5·23-s + 249.·25-s + 89.8·29-s − 183.·31-s + 93.6·35-s + 418.·37-s + 142.·41-s + 71.0·43-s + 323.·47-s − 319.·49-s + 25.1·53-s + 1.18e3·55-s − 684.·59-s + 308.·61-s − 251.·65-s − 672.·67-s − 326.·71-s + 24.3·73-s + 295.·77-s − 166.·79-s − 201.·83-s + ⋯
L(s)  = 1  − 1.72·5-s − 0.261·7-s − 1.67·11-s + 0.277·13-s + 0.596·17-s + 1.29·19-s + 0.258·23-s + 1.99·25-s + 0.575·29-s − 1.06·31-s + 0.452·35-s + 1.85·37-s + 0.543·41-s + 0.252·43-s + 1.00·47-s − 0.931·49-s + 0.0650·53-s + 2.89·55-s − 1.51·59-s + 0.646·61-s − 0.479·65-s − 1.22·67-s − 0.546·71-s + 0.0389·73-s + 0.437·77-s − 0.237·79-s − 0.265·83-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 110.451110.451
Root analytic conductor: 10.509510.5095
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1872, ( :3/2), −1)(2,\ 1872,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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)
bad2 1 1
3 1 1
13 1−13T 1 - 13T
good5 1+19.3T+125T2 1 + 19.3T + 125T^{2}
7 1+4.84T+343T2 1 + 4.84T + 343T^{2}
11 1+61.0T+1.33e3T2 1 + 61.0T + 1.33e3T^{2}
17 1−41.7T+4.91e3T2 1 - 41.7T + 4.91e3T^{2}
19 1−107.T+6.85e3T2 1 - 107.T + 6.85e3T^{2}
23 1−28.5T+1.21e4T2 1 - 28.5T + 1.21e4T^{2}
29 1−89.8T+2.43e4T2 1 - 89.8T + 2.43e4T^{2}
31 1+183.T+2.97e4T2 1 + 183.T + 2.97e4T^{2}
37 1−418.T+5.06e4T2 1 - 418.T + 5.06e4T^{2}
41 1−142.T+6.89e4T2 1 - 142.T + 6.89e4T^{2}
43 1−71.0T+7.95e4T2 1 - 71.0T + 7.95e4T^{2}
47 1−323.T+1.03e5T2 1 - 323.T + 1.03e5T^{2}
53 1−25.1T+1.48e5T2 1 - 25.1T + 1.48e5T^{2}
59 1+684.T+2.05e5T2 1 + 684.T + 2.05e5T^{2}
61 1−308.T+2.26e5T2 1 - 308.T + 2.26e5T^{2}
67 1+672.T+3.00e5T2 1 + 672.T + 3.00e5T^{2}
71 1+326.T+3.57e5T2 1 + 326.T + 3.57e5T^{2}
73 1−24.3T+3.89e5T2 1 - 24.3T + 3.89e5T^{2}
79 1+166.T+4.93e5T2 1 + 166.T + 4.93e5T^{2}
83 1+201.T+5.71e5T2 1 + 201.T + 5.71e5T^{2}
89 1+108.T+7.04e5T2 1 + 108.T + 7.04e5T^{2}
97 1−1.15e3T+9.12e5T2 1 - 1.15e3T + 9.12e5T^{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

−8.245736428002200750136047556038, −7.57547644245008626663418372191, −7.38208841751880378861147131901, −6.01279702449559026804490910220, −5.12714336593394908950690043799, −4.32047385002478398692844630940, −3.33745089719184438097433035452, −2.75545666297865441718368239673, −0.958062341644736310805027065430, 0, 0.958062341644736310805027065430, 2.75545666297865441718368239673, 3.33745089719184438097433035452, 4.32047385002478398692844630940, 5.12714336593394908950690043799, 6.01279702449559026804490910220, 7.38208841751880378861147131901, 7.57547644245008626663418372191, 8.245736428002200750136047556038

Graph of the ZZ-function along the critical line