↑

Properties

Label 2-39e2-1.1-c3-0-147
Degree 22
Conductor 15211521
Sign −1-1
Analytic cond. 89.741989.7419
Root an. cond. 9.473229.47322
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4-s − 9·5-s − 2·7-s − 21·8-s − 27·10-s + 30·11-s − 6·14-s − 71·16-s + 111·17-s + 46·19-s − 9·20-s + 90·22-s + 6·23-s − 44·25-s − 2·28-s + 105·29-s + 100·31-s − 45·32-s + 333·34-s + 18·35-s − 17·37-s + 138·38-s + 189·40-s − 231·41-s − 514·43-s + 30·44-s + ⋯
L(s)  = 1  + 1.06·2-s + 1/8·4-s − 0.804·5-s − 0.107·7-s − 0.928·8-s − 0.853·10-s + 0.822·11-s − 0.114·14-s − 1.10·16-s + 1.58·17-s + 0.555·19-s − 0.100·20-s + 0.872·22-s + 0.0543·23-s − 0.351·25-s − 0.0134·28-s + 0.672·29-s + 0.579·31-s − 0.248·32-s + 1.67·34-s + 0.0869·35-s − 0.0755·37-s + 0.589·38-s + 0.747·40-s − 0.879·41-s − 1.82·43-s + 0.102·44-s + ⋯

Functional equation

Λ(s)=(1521s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1521s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15211521    =    32⋅1323^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 89.741989.7419
Root analytic conductor: 9.473229.47322
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1521, ( :3/2), −1)(2,\ 1521,\ (\ :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)
bad3 1 1
13 1 1
good2 1−3T+p3T2 1 - 3 T + p^{3} T^{2}
5 1+9T+p3T2 1 + 9 T + p^{3} T^{2}
7 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
11 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
17 1−111T+p3T2 1 - 111 T + p^{3} T^{2}
19 1−46T+p3T2 1 - 46 T + p^{3} T^{2}
23 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
29 1−105T+p3T2 1 - 105 T + p^{3} T^{2}
31 1−100T+p3T2 1 - 100 T + p^{3} T^{2}
37 1+17T+p3T2 1 + 17 T + p^{3} T^{2}
41 1+231T+p3T2 1 + 231 T + p^{3} T^{2}
43 1+514T+p3T2 1 + 514 T + p^{3} T^{2}
47 1+162T+p3T2 1 + 162 T + p^{3} T^{2}
53 1+639T+p3T2 1 + 639 T + p^{3} T^{2}
59 1−600T+p3T2 1 - 600 T + p^{3} T^{2}
61 1−233T+p3T2 1 - 233 T + p^{3} T^{2}
67 1+926T+p3T2 1 + 926 T + p^{3} T^{2}
71 1+930T+p3T2 1 + 930 T + p^{3} T^{2}
73 1−253T+p3T2 1 - 253 T + p^{3} T^{2}
79 1+1324T+p3T2 1 + 1324 T + p^{3} T^{2}
83 1−810T+p3T2 1 - 810 T + p^{3} T^{2}
89 1−498T+p3T2 1 - 498 T + p^{3} T^{2}
97 1+14pT+p3T2 1 + 14 p T + p^{3} T^{2}
show more
show less
   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

−8.568338841165216619895259235486, −7.905920592436757601726033674712, −6.89290375199929068953503225010, −6.11538744845931173383891097494, −5.20708260822398427193114217223, −4.46423251969212884473115761847, −3.52476243041538322429584076889, −3.09992476495109335889185740232, −1.35922757941284379572215283487, 0, 1.35922757941284379572215283487, 3.09992476495109335889185740232, 3.52476243041538322429584076889, 4.46423251969212884473115761847, 5.20708260822398427193114217223, 6.11538744845931173383891097494, 6.89290375199929068953503225010, 7.905920592436757601726033674712, 8.568338841165216619895259235486

Graph of the ZZ-function along the critical line