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Properties

Label 8-1152e4-1.1-c3e4-0-3
Degree 88
Conductor 1.761×10121.761\times 10^{12}
Sign 11
Analytic cond. 2.13439×1072.13439\times 10^{7}
Root an. cond. 8.244408.24440
Motivic weight 33
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  + 136·17-s + 484·25-s − 104·41-s − 972·49-s − 1.35e3·73-s + 936·89-s − 712·97-s − 5.51e3·113-s + 4.52e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.65e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 1.94·17-s + 3.87·25-s − 0.396·41-s − 2.83·49-s − 2.16·73-s + 1.11·89-s − 0.745·97-s − 4.58·113-s + 3.39·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.57·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯

Functional equation

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

Invariants

Degree: 88
Conductor: 228⋅382^{28} \cdot 3^{8}
Sign: 11
Analytic conductor: 2.13439×1072.13439\times 10^{7}
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅38, ( :3/2,3/2,3/2,3/2), 1)(8,\ 2^{28} \cdot 3^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 1.2728426621.272842662
L(12)L(\frac12) ≈\approx 1.2728426621.272842662
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
good5C22C_2^2 (1−242T2+p6T4)2 ( 1 - 242 T^{2} + p^{6} T^{4} )^{2}
7C22C_2^2 (1+486T2+p6T4)2 ( 1 + 486 T^{2} + p^{6} T^{4} )^{2}
11C22C_2^2 (1−2262T2+p6T4)2 ( 1 - 2262 T^{2} + p^{6} T^{4} )^{2}
13C22C_2^2 (1−2826T2+p6T4)2 ( 1 - 2826 T^{2} + p^{6} T^{4} )^{2}
17C2C_2 (1−2pT+p3T2)4 ( 1 - 2 p T + p^{3} T^{2} )^{4}
19C22C_2^2 (1−11014T2+p6T4)2 ( 1 - 11014 T^{2} + p^{6} T^{4} )^{2}
23C22C_2^2 (1+20462T2+p6T4)2 ( 1 + 20462 T^{2} + p^{6} T^{4} )^{2}
29C22C_2^2 (1−8450T2+p6T4)2 ( 1 - 8450 T^{2} + p^{6} T^{4} )^{2}
31C22C_2^2 (1+47414T2+p6T4)2 ( 1 + 47414 T^{2} + p^{6} T^{4} )^{2}
37C22C_2^2 (1−27578T2+p6T4)2 ( 1 - 27578 T^{2} + p^{6} T^{4} )^{2}
41C2C_2 (1+26T+p3T2)4 ( 1 + 26 T + p^{3} T^{2} )^{4}
43C22C_2^2 (1−95510T2+p6T4)2 ( 1 - 95510 T^{2} + p^{6} T^{4} )^{2}
47C22C_2^2 (1+88574T2+p6T4)2 ( 1 + 88574 T^{2} + p^{6} T^{4} )^{2}
53C22C_2^2 (1+166894T2+p6T4)2 ( 1 + 166894 T^{2} + p^{6} T^{4} )^{2}
59C22C_2^2 (1−278262T2+p6T4)2 ( 1 - 278262 T^{2} + p^{6} T^{4} )^{2}
61C22C_2^2 (1+86838T2+p6T4)2 ( 1 + 86838 T^{2} + p^{6} T^{4} )^{2}
67C22C_2^2 (1−207142T2+p6T4)2 ( 1 - 207142 T^{2} + p^{6} T^{4} )^{2}
71C22C_2^2 (1+604430T2+p6T4)2 ( 1 + 604430 T^{2} + p^{6} T^{4} )^{2}
73C2C_2 (1+338T+p3T2)4 ( 1 + 338 T + p^{3} T^{2} )^{4}
79C22C_2^2 (1+363350T2+p6T4)2 ( 1 + 363350 T^{2} + p^{6} T^{4} )^{2}
83C22C_2^2 (1−70278T2+p6T4)2 ( 1 - 70278 T^{2} + p^{6} T^{4} )^{2}
89C2C_2 (1−234T+p3T2)4 ( 1 - 234 T + p^{3} T^{2} )^{4}
97C2C_2 (1+178T+p3T2)4 ( 1 + 178 T + p^{3} T^{2} )^{4}
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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

−6.63135501773375096995467674486, −6.50262763593968272060045176338, −6.14984553361220615793207428036, −5.88406683967119816162935653518, −5.68787735160044874478662116797, −5.57700880814428976868814636181, −5.16276399423517679026567943673, −5.05105394392340921228318370694, −4.80639317263846726247507254275, −4.61542322328332252282337477321, −4.28906949436810009805523582003, −4.28556533417888526557360441913, −3.64761301963849003627339300308, −3.45421463787945058567909111710, −3.16850331101589513900898739075, −3.12214587073399288990465150740, −2.86893437219061693048737501733, −2.67386613528431030394105818079, −2.02914152183182118196556125037, −1.92922015367171837719337752878, −1.44398644820311621684666066920, −1.24064297640487049660214893673, −0.847554099171917711023957890501, −0.796901520061660001420263519702, −0.11922734409271183735413540236, 0.11922734409271183735413540236, 0.796901520061660001420263519702, 0.847554099171917711023957890501, 1.24064297640487049660214893673, 1.44398644820311621684666066920, 1.92922015367171837719337752878, 2.02914152183182118196556125037, 2.67386613528431030394105818079, 2.86893437219061693048737501733, 3.12214587073399288990465150740, 3.16850331101589513900898739075, 3.45421463787945058567909111710, 3.64761301963849003627339300308, 4.28556533417888526557360441913, 4.28906949436810009805523582003, 4.61542322328332252282337477321, 4.80639317263846726247507254275, 5.05105394392340921228318370694, 5.16276399423517679026567943673, 5.57700880814428976868814636181, 5.68787735160044874478662116797, 5.88406683967119816162935653518, 6.14984553361220615793207428036, 6.50262763593968272060045176338, 6.63135501773375096995467674486

Graph of the ZZ-function along the critical line