↑

Properties

Label 2-912-1.1-c3-0-11
Degree 22
Conductor 912912
Sign 11
Analytic cond. 53.809753.8097
Root an. cond. 7.335517.33551
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 7·5-s + 15·7-s + 9·9-s + 49·11-s + 14·13-s + 21·15-s − 33·17-s + 19·19-s − 45·21-s + 148·23-s − 76·25-s − 27·27-s − 278·29-s − 94·31-s − 147·33-s − 105·35-s + 160·37-s − 42·39-s + 400·41-s − 73·43-s − 63·45-s − 173·47-s − 118·49-s + 99·51-s + 170·53-s − 343·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.626·5-s + 0.809·7-s + 1/3·9-s + 1.34·11-s + 0.298·13-s + 0.361·15-s − 0.470·17-s + 0.229·19-s − 0.467·21-s + 1.34·23-s − 0.607·25-s − 0.192·27-s − 1.78·29-s − 0.544·31-s − 0.775·33-s − 0.507·35-s + 0.710·37-s − 0.172·39-s + 1.52·41-s − 0.258·43-s − 0.208·45-s − 0.536·47-s − 0.344·49-s + 0.271·51-s + 0.440·53-s − 0.840·55-s + ⋯

Functional equation

Λ(s)=(912s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(912s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 912912    =    24⋅3⋅192^{4} \cdot 3 \cdot 19
Sign: 11
Analytic conductor: 53.809753.8097
Root analytic conductor: 7.335517.33551
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 912, ( :3/2), 1)(2,\ 912,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.7399249051.739924905
L(12)L(\frac12) ≈\approx 1.7399249051.739924905
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)
bad2 1 1
3 1+pT 1 + p T
19 1−pT 1 - p T
good5 1+7T+p3T2 1 + 7 T + p^{3} T^{2}
7 1−15T+p3T2 1 - 15 T + p^{3} T^{2}
11 1−49T+p3T2 1 - 49 T + p^{3} T^{2}
13 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
17 1+33T+p3T2 1 + 33 T + p^{3} T^{2}
23 1−148T+p3T2 1 - 148 T + p^{3} T^{2}
29 1+278T+p3T2 1 + 278 T + p^{3} T^{2}
31 1+94T+p3T2 1 + 94 T + p^{3} T^{2}
37 1−160T+p3T2 1 - 160 T + p^{3} T^{2}
41 1−400T+p3T2 1 - 400 T + p^{3} T^{2}
43 1+73T+p3T2 1 + 73 T + p^{3} T^{2}
47 1+173T+p3T2 1 + 173 T + p^{3} T^{2}
53 1−170T+p3T2 1 - 170 T + p^{3} T^{2}
59 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
61 1−419T+p3T2 1 - 419 T + p^{3} T^{2}
67 1+444T+p3T2 1 + 444 T + p^{3} T^{2}
71 1−952T+p3T2 1 - 952 T + p^{3} T^{2}
73 1+27T+p3T2 1 + 27 T + p^{3} T^{2}
79 1−556T+p3T2 1 - 556 T + p^{3} T^{2}
83 1−276T+p3T2 1 - 276 T + p^{3} T^{2}
89 1−1386T+p3T2 1 - 1386 T + p^{3} T^{2}
97 1−130T+p3T2 1 - 130 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

−9.556392717028457585239498153050, −8.973340666111262649883877927079, −7.904685113893163395493555917809, −7.18490926444383802894774417904, −6.27754130620658587456707528300, −5.29311764947051825021823018017, −4.32533145980041930446190233461, −3.58427192136092151025726774775, −1.87355512482263125898090653131, −0.77043222100486511123292230306, 0.77043222100486511123292230306, 1.87355512482263125898090653131, 3.58427192136092151025726774775, 4.32533145980041930446190233461, 5.29311764947051825021823018017, 6.27754130620658587456707528300, 7.18490926444383802894774417904, 7.904685113893163395493555917809, 8.973340666111262649883877927079, 9.556392717028457585239498153050

Graph of the ZZ-function along the critical line