↑

Properties

Label 2-15456-1.1-c1-0-16
Degree 22
Conductor 1545615456
Sign −1-1
Analytic cond. 123.416123.416
Root an. cond. 11.109311.1093
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 7-s + 9-s − 2·15-s + 2·19-s + 21-s + 23-s − 25-s + 27-s + 6·29-s − 6·31-s − 2·35-s − 6·37-s − 2·41-s − 2·45-s − 6·47-s + 49-s − 10·53-s + 2·57-s + 12·59-s + 2·61-s + 63-s − 8·67-s + 69-s + 4·71-s + 14·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.516·15-s + 0.458·19-s + 0.218·21-s + 0.208·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.07·31-s − 0.338·35-s − 0.986·37-s − 0.312·41-s − 0.298·45-s − 0.875·47-s + 1/7·49-s − 1.37·53-s + 0.264·57-s + 1.56·59-s + 0.256·61-s + 0.125·63-s − 0.977·67-s + 0.120·69-s + 0.474·71-s + 1.63·73-s + ⋯

Functional equation

Λ(s)=(15456s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 15456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(15456s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1545615456    =    25⋅3⋅7⋅232^{5} \cdot 3 \cdot 7 \cdot 23
Sign: −1-1
Analytic conductor: 123.416123.416
Root analytic conductor: 11.109311.1093
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 15456, ( :1/2), −1)(2,\ 15456,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
7 1−T 1 - T
23 1−T 1 - T
good5 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+6T+pT2 1 + 6 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1+8T+pT2 1 + 8 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

−16.10555942276634, −15.74274816560423, −15.19122966434810, −14.60858356299189, −14.19055018182533, −13.56509180900508, −12.98199863815776, −12.28902257082646, −11.88024575898632, −11.23599608619931, −10.73269056655203, −9.994586645592473, −9.422152955207514, −8.728868051909036, −8.166945512697150, −7.794467891403980, −7.051434037027880, −6.589969506622638, −5.533048697322881, −4.956227228978669, −4.208379786229513, −3.591079860284131, −2.990612346941021, −2.051625024283689, −1.196766132841192, 0, 1.196766132841192, 2.051625024283689, 2.990612346941021, 3.591079860284131, 4.208379786229513, 4.956227228978669, 5.533048697322881, 6.589969506622638, 7.051434037027880, 7.794467891403980, 8.166945512697150, 8.728868051909036, 9.422152955207514, 9.994586645592473, 10.73269056655203, 11.23599608619931, 11.88024575898632, 12.28902257082646, 12.98199863815776, 13.56509180900508, 14.19055018182533, 14.60858356299189, 15.19122966434810, 15.74274816560423, 16.10555942276634

Graph of the ZZ-function along the critical line