↑

Properties

Label 2-1472-1.1-c1-0-12
Degree 22
Conductor 14721472
Sign 11
Analytic cond. 11.753911.7539
Root an. cond. 3.428403.42840
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 3·9-s − 6·11-s + 2·13-s + 6·17-s + 6·19-s + 23-s − 5·25-s + 6·29-s + 8·37-s + 6·41-s + 2·43-s − 8·47-s + 9·49-s + 8·53-s − 4·59-s + 4·61-s − 12·63-s − 2·67-s − 8·71-s + 6·73-s − 24·77-s + 12·79-s + 9·81-s − 10·83-s + 10·89-s + 8·91-s + ⋯
L(s)  = 1  + 1.51·7-s − 9-s − 1.80·11-s + 0.554·13-s + 1.45·17-s + 1.37·19-s + 0.208·23-s − 25-s + 1.11·29-s + 1.31·37-s + 0.937·41-s + 0.304·43-s − 1.16·47-s + 9/7·49-s + 1.09·53-s − 0.520·59-s + 0.512·61-s − 1.51·63-s − 0.244·67-s − 0.949·71-s + 0.702·73-s − 2.73·77-s + 1.35·79-s + 81-s − 1.09·83-s + 1.05·89-s + 0.838·91-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: 11
Analytic conductor: 11.753911.7539
Root analytic conductor: 3.428403.42840
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1472, ( :1/2), 1)(2,\ 1472,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8370277731.837027773
L(12)L(\frac12) ≈\approx 1.8370277731.837027773
L(32)L(\frac{3}{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
23 1−T 1 - T
good3 1+pT2 1 + p T^{2}
5 1+pT2 1 + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−8T+pT2 1 - 8 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−4T+pT2 1 - 4 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−12T+pT2 1 - 12 T + p T^{2}
83 1+10T+pT2 1 + 10 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+18T+pT2 1 + 18 T + p 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

−9.558010235892985284674615381616, −8.374509306974827036411232822607, −7.964503400926076755155618669838, −7.48215223824333324077746116110, −5.86176128573934039100170052811, −5.42499451908998134631580290849, −4.67644498964062625267561415914, −3.28949124296187423948746764298, −2.42562828697154691354778403293, −1.01213231694948405739618182908, 1.01213231694948405739618182908, 2.42562828697154691354778403293, 3.28949124296187423948746764298, 4.67644498964062625267561415914, 5.42499451908998134631580290849, 5.86176128573934039100170052811, 7.48215223824333324077746116110, 7.964503400926076755155618669838, 8.374509306974827036411232822607, 9.558010235892985284674615381616

Graph of the ZZ-function along the critical line