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Properties

Label 8-2300e4-1.1-c1e4-0-0
Degree 88
Conductor 2.798×10132.798\times 10^{13}
Sign 11
Analytic cond. 113767.113767.
Root an. cond. 4.285504.28550
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·9-s + 8·11-s − 24·19-s − 8·29-s − 4·31-s + 19·49-s − 10·59-s − 12·61-s + 24·71-s − 36·79-s − 7·81-s − 24·89-s + 24·99-s − 46·101-s − 12·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 39·169-s − 72·171-s + ⋯
L(s)  = 1  + 9-s + 2.41·11-s − 5.50·19-s − 1.48·29-s − 0.718·31-s + 19/7·49-s − 1.30·59-s − 1.53·61-s + 2.84·71-s − 4.05·79-s − 7/9·81-s − 2.54·89-s + 2.41·99-s − 4.57·101-s − 1.14·109-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3·169-s − 5.50·171-s + ⋯

Functional equation

Λ(s)=((28⋅58⋅234)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((28⋅58⋅234)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28⋅58⋅2342^{8} \cdot 5^{8} \cdot 23^{4}
Sign: 11
Analytic conductor: 113767.113767.
Root analytic conductor: 4.285504.28550
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅58⋅234, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{8} \cdot 5^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.2544503332.254450333
L(12)L(\frac12) ≈\approx 2.2544503332.254450333
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
5 1 1
23C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good3D4×C2D_4\times C_2 1−pT2+16T4−p3T6+p4T8 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8}
7D4×C2D_4\times C_2 1−19T2+184T4−19p2T6+p4T8 1 - 19 T^{2} + 184 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8}
11C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
13D4×C2D_4\times C_2 1−3pT2+680T4−3p3T6+p4T8 1 - 3 p T^{2} + 680 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8}
17D4×C2D_4\times C_2 1−59T2+1444T4−59p2T6+p4T8 1 - 59 T^{2} + 1444 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8}
19C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
29D4D_{4} (1+4T+45T2+4pT3+p2T4)2 ( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
31D4D_{4} (1+2T−5T2+2pT3+p2T4)2 ( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1−127T2+6664T4−127p2T6+p4T8 1 - 127 T^{2} + 6664 T^{4} - 127 p^{2} T^{6} + p^{4} T^{8}
41C22C_2^2 (1+65T2+p2T4)2 ( 1 + 65 T^{2} + p^{2} T^{4} )^{2}
43C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
47D4×C2D_4\times C_2 1−107T2+6936T4−107p2T6+p4T8 1 - 107 T^{2} + 6936 T^{4} - 107 p^{2} T^{6} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−191T2+14632T4−191p2T6+p4T8 1 - 191 T^{2} + 14632 T^{4} - 191 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1+5T+86T2+5pT3+p2T4)2 ( 1 + 5 T + 86 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2}
61D4D_{4} (1+6T+114T2+6pT3+p2T4)2 ( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−175T2+15916T4−175p2T6+p4T8 1 - 175 T^{2} + 15916 T^{4} - 175 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1−12T+161T2−12pT3+p2T4)2 ( 1 - 12 T + 161 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1−199T2+19840T4−199p2T6+p4T8 1 - 199 T^{2} + 19840 T^{4} - 199 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1+18T+222T2+18pT3+p2T4)2 ( 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1+85T2+15376T4+85p2T6+p4T8 1 + 85 T^{2} + 15376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8}
89D4D_{4} (1+12T+146T2+12pT3+p2T4)2 ( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−192T2+22526T4−192p2T6+p4T8 1 - 192 T^{2} + 22526 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8}
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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

−6.47720705641153756255839716199, −6.20157648963508884697311454059, −6.12349114470830002877418782494, −5.69174898879537983679204803186, −5.66242896332010733462295868651, −5.43466003893453319520443709987, −5.34506170508296469213931086513, −4.54208524705766411609994900361, −4.50418075742600502692610010654, −4.49639262794295976712704068816, −4.26527282186323017147359551419, −4.00428535033584133867341723346, −3.94674267463688639654017916825, −3.83383714250876091929595743266, −3.54952314532672067303804848579, −3.02932428616861243646637013049, −2.65485863579735171951860026419, −2.62845434297040578089079416770, −2.32894006910848013716554516161, −1.81272631460850310706014414490, −1.64604722216838827234324406590, −1.51112087306479009439007422655, −1.49522588001476367536538684544, −0.52818152986298899885087905296, −0.32703884585831648308667288333, 0.32703884585831648308667288333, 0.52818152986298899885087905296, 1.49522588001476367536538684544, 1.51112087306479009439007422655, 1.64604722216838827234324406590, 1.81272631460850310706014414490, 2.32894006910848013716554516161, 2.62845434297040578089079416770, 2.65485863579735171951860026419, 3.02932428616861243646637013049, 3.54952314532672067303804848579, 3.83383714250876091929595743266, 3.94674267463688639654017916825, 4.00428535033584133867341723346, 4.26527282186323017147359551419, 4.49639262794295976712704068816, 4.50418075742600502692610010654, 4.54208524705766411609994900361, 5.34506170508296469213931086513, 5.43466003893453319520443709987, 5.66242896332010733462295868651, 5.69174898879537983679204803186, 6.12349114470830002877418782494, 6.20157648963508884697311454059, 6.47720705641153756255839716199

Graph of the ZZ-function along the critical line