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Properties

Label 8-2160e4-1.1-c2e4-0-1
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 1.19992×1071.19992\times 10^{7}
Root an. cond. 7.671747.67174
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  − 20·7-s − 28·13-s + 4·19-s − 10·25-s + 28·31-s − 64·37-s − 116·43-s + 144·49-s − 4·61-s − 56·67-s − 28·73-s + 268·79-s + 560·91-s + 8·97-s + 400·103-s + 140·109-s + 376·121-s + 127-s + 131-s − 80·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2.85·7-s − 2.15·13-s + 4/19·19-s − 2/5·25-s + 0.903·31-s − 1.72·37-s − 2.69·43-s + 2.93·49-s − 0.0655·61-s − 0.835·67-s − 0.383·73-s + 3.39·79-s + 6.15·91-s + 8/97·97-s + 3.88·103-s + 1.28·109-s + 3.10·121-s + 0.00787·127-s + 0.00763·131-s − 0.601·133-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)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅312⋅542^{16} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 1.19992×1071.19992\times 10^{7}
Root analytic conductor: 7.671747.67174
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1,1,1,1), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.26488109440.2648810944
L(12)L(\frac12) ≈\approx 0.26488109440.2648810944
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
5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
good7D4D_{4} (1+10T+78T2+10p2T3+p4T4)2 ( 1 + 10 T + 78 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2}
11D4×C2D_4\times C_2 1−376T2+63006T4−376p4T6+p8T8 1 - 376 T^{2} + 63006 T^{4} - 376 p^{4} T^{6} + p^{8} T^{8}
13D4D_{4} (1+14T+342T2+14p2T3+p4T4)2 ( 1 + 14 T + 342 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2}
17C22C_2^2 (1−353T2+p4T4)2 ( 1 - 353 T^{2} + p^{4} T^{4} )^{2}
19D4D_{4} (1−2T+543T2−2p2T3+p4T4)2 ( 1 - 2 T + 543 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2}
23D4×C2D_4\times C_2 1−38pT2+433131T4−38p5T6+p8T8 1 - 38 p T^{2} + 433131 T^{4} - 38 p^{5} T^{6} + p^{8} T^{8}
29D4×C2D_4\times C_2 1−3256T2+4063326T4−3256p4T6+p8T8 1 - 3256 T^{2} + 4063326 T^{4} - 3256 p^{4} T^{6} + p^{8} T^{8}
31D4D_{4} (1−14T+1791T2−14p2T3+p4T4)2 ( 1 - 14 T + 1791 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2}
37D4D_{4} (1+32T+1374T2+32p2T3+p4T4)2 ( 1 + 32 T + 1374 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2}
41D4×C2D_4\times C_2 1−4312T2+9935358T4−4312p4T6+p8T8 1 - 4312 T^{2} + 9935358 T^{4} - 4312 p^{4} T^{6} + p^{8} T^{8}
43D4D_{4} (1+58T+2334T2+58p2T3+p4T4)2 ( 1 + 58 T + 2334 T^{2} + 58 p^{2} T^{3} + p^{4} T^{4} )^{2}
47D4×C2D_4\times C_2 1−8188T2+26416518T4−8188p4T6+p8T8 1 - 8188 T^{2} + 26416518 T^{4} - 8188 p^{4} T^{6} + p^{8} T^{8}
53D4×C2D_4\times C_2 1−2218T2+16848843T4−2218p4T6+p8T8 1 - 2218 T^{2} + 16848843 T^{4} - 2218 p^{4} T^{6} + p^{8} T^{8}
59D4×C2D_4\times C_2 1−7912T2+35671038T4−7912p4T6+p8T8 1 - 7912 T^{2} + 35671038 T^{4} - 7912 p^{4} T^{6} + p^{8} T^{8}
61D4D_{4} (1+2T+963T2+2p2T3+p4T4)2 ( 1 + 2 T + 963 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2}
67C2C_2 (1+14T+p2T2)4 ( 1 + 14 T + p^{2} T^{2} )^{4}
71D4×C2D_4\times C_2 1−11704T2+67208766T4−11704p4T6+p8T8 1 - 11704 T^{2} + 67208766 T^{4} - 11704 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1+14T+10302T2+14p2T3+p4T4)2 ( 1 + 14 T + 10302 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2}
79D4D_{4} (1−134T+10491T2−134p2T3+p4T4)2 ( 1 - 134 T + 10491 T^{2} - 134 p^{2} T^{3} + p^{4} T^{4} )^{2}
83D4×C2D_4\times C_2 1−2338T2+46102083T4−2338p4T6+p8T8 1 - 2338 T^{2} + 46102083 T^{4} - 2338 p^{4} T^{6} + p^{8} T^{8}
89D4×C2D_4\times C_2 1−5512T2+128866398T4−5512p4T6+p8T8 1 - 5512 T^{2} + 128866398 T^{4} - 5512 p^{4} T^{6} + p^{8} T^{8}
97D4D_{4} (1−4T+18642T2−4p2T3+p4T4)2 ( 1 - 4 T + 18642 T^{2} - 4 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

−6.15520337940245998411410790506, −6.11975227575004077631893119551, −5.95303973762374004943388819176, −5.93844323699500108986644232054, −5.30132916203829281385766317978, −5.06225181883497336214245021633, −4.96173449225333657528992891265, −4.80772815357307259725800855289, −4.74801411947112610582821577092, −4.47551953854594537920016044272, −3.92028767413694705581123786683, −3.64370874083565214362974430099, −3.48761474003375215977065483819, −3.45209848100669186490158424148, −3.36019647639628026039147381361, −2.95447763617178436020835607191, −2.55541853993553361180115240354, −2.51438944098647962286301159526, −2.25443585167870867333135640754, −1.96931647892790466040500233570, −1.49956988696577790762575541113, −1.33078258958841618537562736342, −0.57920948154722457338545198220, −0.48805469851330721458364115266, −0.11225567927891783844417251355, 0.11225567927891783844417251355, 0.48805469851330721458364115266, 0.57920948154722457338545198220, 1.33078258958841618537562736342, 1.49956988696577790762575541113, 1.96931647892790466040500233570, 2.25443585167870867333135640754, 2.51438944098647962286301159526, 2.55541853993553361180115240354, 2.95447763617178436020835607191, 3.36019647639628026039147381361, 3.45209848100669186490158424148, 3.48761474003375215977065483819, 3.64370874083565214362974430099, 3.92028767413694705581123786683, 4.47551953854594537920016044272, 4.74801411947112610582821577092, 4.80772815357307259725800855289, 4.96173449225333657528992891265, 5.06225181883497336214245021633, 5.30132916203829281385766317978, 5.93844323699500108986644232054, 5.95303973762374004943388819176, 6.11975227575004077631893119551, 6.15520337940245998411410790506

Graph of the ZZ-function along the critical line