↑

Properties

Label 2-304920-1.1-c1-0-68
Degree 22
Conductor 304920304920
Sign 11
Analytic cond. 2434.792434.79
Root an. cond. 49.343649.3436
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s + 6·13-s − 2·17-s + 8·19-s − 4·23-s + 25-s + 6·29-s + 4·31-s + 35-s − 2·37-s − 2·41-s + 12·43-s + 49-s − 2·53-s + 4·59-s − 6·61-s + 6·65-s − 4·67-s − 8·71-s − 6·73-s + 16·79-s − 4·83-s − 2·85-s + 18·89-s + 6·91-s + 8·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.66·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.169·35-s − 0.328·37-s − 0.312·41-s + 1.82·43-s + 1/7·49-s − 0.274·53-s + 0.520·59-s − 0.768·61-s + 0.744·65-s − 0.488·67-s − 0.949·71-s − 0.702·73-s + 1.80·79-s − 0.439·83-s − 0.216·85-s + 1.90·89-s + 0.628·91-s + 0.820·95-s + ⋯

Functional equation

Λ(s)=(304920s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 304920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(304920s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 304920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 304920304920    =    23⋅32⋅5⋅7⋅1122^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11^{2}
Sign: 11
Analytic conductor: 2434.792434.79
Root analytic conductor: 49.343649.3436
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 304920, ( :1/2), 1)(2,\ 304920,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.5888577584.588857758
L(12)L(\frac12) ≈\approx 4.5888577584.588857758
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
3 1 1
5 1−T 1 - T
7 1−T 1 - T
11 1 1
good13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+4T+pT2 1 + 4 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 1−16T+pT2 1 - 16 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−18T+pT2 1 - 18 T + p T^{2}
97 1−6T+pT2 1 - 6 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

−12.62417825330028, −12.04910305277411, −11.88502112956148, −11.24915489137420, −10.86832381911353, −10.38720919889787, −10.02120589339974, −9.339756129753913, −9.053545539475780, −8.538681513389040, −8.063764576924764, −7.617293432889393, −7.110345394671678, −6.408117143357007, −6.114062486233130, −5.695701137972401, −5.055333646088013, −4.631980049837234, −3.976742269605177, −3.523297852218326, −2.910761246729892, −2.395930473741585, −1.605687474779605, −1.172701737115260, −0.6211932209128952, 0.6211932209128952, 1.172701737115260, 1.605687474779605, 2.395930473741585, 2.910761246729892, 3.523297852218326, 3.976742269605177, 4.631980049837234, 5.055333646088013, 5.695701137972401, 6.114062486233130, 6.408117143357007, 7.110345394671678, 7.617293432889393, 8.063764576924764, 8.538681513389040, 9.053545539475780, 9.339756129753913, 10.02120589339974, 10.38720919889787, 10.86832381911353, 11.24915489137420, 11.88502112956148, 12.04910305277411, 12.62417825330028

Graph of the ZZ-function along the critical line