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Properties

Label 8-1152e4-1.1-c2e4-0-12
Degree 88
Conductor 1.761×10121.761\times 10^{12}
Sign 11
Analytic cond. 970845.970845.
Root an. cond. 5.602655.60265
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s − 32·13-s + 32·19-s + 72·25-s + 120·31-s − 88·37-s + 144·43-s + 36·49-s − 200·61-s + 112·67-s + 272·73-s + 488·79-s + 256·91-s + 160·97-s + 40·103-s − 192·109-s + 372·121-s + 127-s + 131-s − 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 8/7·7-s − 2.46·13-s + 1.68·19-s + 2.87·25-s + 3.87·31-s − 2.37·37-s + 3.34·43-s + 0.734·49-s − 3.27·61-s + 1.67·67-s + 3.72·73-s + 6.17·79-s + 2.81·91-s + 1.64·97-s + 0.388·103-s − 1.76·109-s + 3.07·121-s + 0.00787·127-s + 0.00763·131-s − 1.92·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯

Functional equation

Λ(s)=((228⋅38)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((228⋅38)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228⋅382^{28} \cdot 3^{8}
Sign: 11
Analytic conductor: 970845.970845.
Root analytic conductor: 5.602655.60265
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅38, ( :1,1,1,1), 1)(8,\ 2^{28} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 5.0289390935.028939093
L(12)L(\frac12) ≈\approx 5.0289390935.028939093
L(2)L(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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
good5D4×C2D_4\times C_2 1−72T2+98p2T4−72p4T6+p8T8 1 - 72 T^{2} + 98 p^{2} T^{4} - 72 p^{4} T^{6} + p^{8} T^{8}
7D4D_{4} (1+4T+6T2+4p2T3+p4T4)2 ( 1 + 4 T + 6 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2}
11D4×C2D_4\times C_2 1−372T2+62342T4−372p4T6+p8T8 1 - 372 T^{2} + 62342 T^{4} - 372 p^{4} T^{6} + p^{8} T^{8}
13D4D_{4} (1+16T+186T2+16p2T3+p4T4)2 ( 1 + 16 T + 186 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2}
17D4×C2D_4\times C_2 1−288T2+184322T4−288p4T6+p8T8 1 - 288 T^{2} + 184322 T^{4} - 288 p^{4} T^{6} + p^{8} T^{8}
19D4D_{4} (1−16T+690T2−16p2T3+p4T4)2 ( 1 - 16 T + 690 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2}
23C22C_2^2 (1−858T2+p4T4)2 ( 1 - 858 T^{2} + p^{4} T^{4} )^{2}
29D4×C2D_4\times C_2 1−2152T2+2530002T4−2152p4T6+p8T8 1 - 2152 T^{2} + 2530002 T^{4} - 2152 p^{4} T^{6} + p^{8} T^{8}
31D4D_{4} (1−60T+2726T2−60p2T3+p4T4)2 ( 1 - 60 T + 2726 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2}
37D4D_{4} (1+44T+2838T2+44p2T3+p4T4)2 ( 1 + 44 T + 2838 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2}
41D4×C2D_4\times C_2 1−2560T2+3056322T4−2560p4T6+p8T8 1 - 2560 T^{2} + 3056322 T^{4} - 2560 p^{4} T^{6} + p^{8} T^{8}
43D4D_{4} (1−72T+3458T2−72p2T3+p4T4)2 ( 1 - 72 T + 3458 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2}
47D4×C2D_4\times C_2 1−6900T2+21047462T4−6900p4T6+p8T8 1 - 6900 T^{2} + 21047462 T^{4} - 6900 p^{4} T^{6} + p^{8} T^{8}
53D4×C2D_4\times C_2 1−7720T2+28930962T4−7720p4T6+p8T8 1 - 7720 T^{2} + 28930962 T^{4} - 7720 p^{4} T^{6} + p^{8} T^{8}
59D4×C2D_4\times C_2 1+1500T2+2678822T4+1500p4T6+p8T8 1 + 1500 T^{2} + 2678822 T^{4} + 1500 p^{4} T^{6} + p^{8} T^{8}
61D4D_{4} (1+100T+8406T2+100p2T3+p4T4)2 ( 1 + 100 T + 8406 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2}
67D4D_{4} (1−56T+9666T2−56p2T3+p4T4)2 ( 1 - 56 T + 9666 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2}
71D4×C2D_4\times C_2 1−5044T2+16873062T4−5044p4T6+p8T8 1 - 5044 T^{2} + 16873062 T^{4} - 5044 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1−136T+14898T2−136p2T3+p4T4)2 ( 1 - 136 T + 14898 T^{2} - 136 p^{2} T^{3} + p^{4} T^{4} )^{2}
79D4D_{4} (1−244T+26502T2−244p2T3+p4T4)2 ( 1 - 244 T + 26502 T^{2} - 244 p^{2} T^{3} + p^{4} T^{4} )^{2}
83D4×C2D_4\times C_2 1+1484T2−19009338T4+1484p4T6+p8T8 1 + 1484 T^{2} - 19009338 T^{4} + 1484 p^{4} T^{6} + p^{8} T^{8}
89D4×C2D_4\times C_2 1−11712T2+138025922T4−11712p4T6+p8T8 1 - 11712 T^{2} + 138025922 T^{4} - 11712 p^{4} T^{6} + p^{8} T^{8}
97D4D_{4} (1−80T+4194T2−80p2T3+p4T4)2 ( 1 - 80 T + 4194 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.88961183828961391015109324318, −6.50223555695454273667829709796, −6.32670867689445088743032657016, −6.27769106380536491595246363123, −6.14208382147177271435000910262, −5.49778010711917415983575565184, −5.47678830966665186965807040883, −5.11811899401695851129424763336, −4.93282626490535246798707245544, −4.79847217933596630385796533873, −4.65400458477585401415392265607, −4.43999388072695122446673076133, −4.05026804549170209153675096445, −3.53875495838113256121120469402, −3.35038757233585790783861980824, −3.24268340789671488726901523034, −3.10426346327897135685863972591, −2.50647276578129634348582102298, −2.48327639166800387397039894368, −2.26671852764699627919093088516, −1.97948683099674901851727037853, −1.12494548331399667867745763690, −0.841463943587056048169589226301, −0.804914850890778932411711137567, −0.40491247981270283916694061408, 0.40491247981270283916694061408, 0.804914850890778932411711137567, 0.841463943587056048169589226301, 1.12494548331399667867745763690, 1.97948683099674901851727037853, 2.26671852764699627919093088516, 2.48327639166800387397039894368, 2.50647276578129634348582102298, 3.10426346327897135685863972591, 3.24268340789671488726901523034, 3.35038757233585790783861980824, 3.53875495838113256121120469402, 4.05026804549170209153675096445, 4.43999388072695122446673076133, 4.65400458477585401415392265607, 4.79847217933596630385796533873, 4.93282626490535246798707245544, 5.11811899401695851129424763336, 5.47678830966665186965807040883, 5.49778010711917415983575565184, 6.14208382147177271435000910262, 6.27769106380536491595246363123, 6.32670867689445088743032657016, 6.50223555695454273667829709796, 6.88961183828961391015109324318

Graph of the ZZ-function along the critical line