↑

Properties

Label 2-4056-1.1-c1-0-18
Degree 22
Conductor 40564056
Sign 11
Analytic cond. 32.387332.3873
Root an. cond. 5.690985.69098
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  + 3-s − 2·5-s + 9-s − 2·15-s + 2·17-s + 4·19-s − 25-s + 27-s + 6·29-s + 2·37-s − 6·41-s − 12·43-s − 2·45-s + 4·47-s − 7·49-s + 2·51-s + 6·53-s + 4·57-s + 8·59-s − 2·61-s − 4·67-s + 12·71-s + 14·73-s − 75-s + 81-s − 8·83-s − 4·85-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.516·15-s + 0.485·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.328·37-s − 0.937·41-s − 1.82·43-s − 0.298·45-s + 0.583·47-s − 49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s + 1.04·59-s − 0.256·61-s − 0.488·67-s + 1.42·71-s + 1.63·73-s − 0.115·75-s + 1/9·81-s − 0.878·83-s − 0.433·85-s + ⋯

Functional equation

Λ(s)=(4056s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4056s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40564056    =    23⋅3⋅1322^{3} \cdot 3 \cdot 13^{2}
Sign: 11
Analytic conductor: 32.387332.3873
Root analytic conductor: 5.690985.69098
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4056, ( :1/2), 1)(2,\ 4056,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9721664201.972166420
L(12)L(\frac12) ≈\approx 1.9721664201.972166420
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−T 1 - T
13 1 1
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+12T+pT2 1 + 12 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+8T+pT2 1 + 8 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

−8.277330555848996337592461167239, −7.85749416685147120585350583537, −7.11646809983531549988243962761, −6.40158815402073049865047684610, −5.30378379001641513519044692133, −4.61671937527489946798974678455, −3.62541556518995470040403604896, −3.19166320207550947955763770245, −2.02833356692568283872392736149, −0.78722055676041492708430177901, 0.78722055676041492708430177901, 2.02833356692568283872392736149, 3.19166320207550947955763770245, 3.62541556518995470040403604896, 4.61671937527489946798974678455, 5.30378379001641513519044692133, 6.40158815402073049865047684610, 7.11646809983531549988243962761, 7.85749416685147120585350583537, 8.277330555848996337592461167239

Graph of the ZZ-function along the critical line