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Properties

Label 8-384e4-1.1-c1e4-0-5
Degree 88
Conductor 2174327193621743271936
Sign 11
Analytic cond. 88.396188.3961
Root an. cond. 1.751071.75107
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  + 2·3-s + 2·9-s − 12·11-s + 8·23-s + 8·25-s + 6·27-s − 24·33-s − 16·37-s + 32·47-s + 16·49-s − 4·59-s + 16·61-s + 16·69-s + 8·71-s − 8·73-s + 16·75-s + 11·81-s − 20·83-s − 16·97-s − 24·99-s + 12·107-s − 16·109-s − 32·111-s + 56·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.15·3-s + 2/3·9-s − 3.61·11-s + 1.66·23-s + 8/5·25-s + 1.15·27-s − 4.17·33-s − 2.63·37-s + 4.66·47-s + 16/7·49-s − 0.520·59-s + 2.04·61-s + 1.92·69-s + 0.949·71-s − 0.936·73-s + 1.84·75-s + 11/9·81-s − 2.19·83-s − 1.62·97-s − 2.41·99-s + 1.16·107-s − 1.53·109-s − 3.03·111-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

Λ(s)=((228⋅34)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((228⋅34)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228⋅342^{28} \cdot 3^{4}
Sign: 11
Analytic conductor: 88.396188.3961
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅34, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.4885754442.488575444
L(12)L(\frac12) ≈\approx 2.4885754442.488575444
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
3C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
good5D4×C2D_4\times C_2 1−8T2+46T4−8p2T6+p4T8 1 - 8 T^{2} + 46 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}
7D4×C2D_4\times C_2 1−16T2+142T4−16p2T6+p4T8 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}
11C4C_4 (1+6T+26T2+6pT3+p2T4)2 ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
13C22C_2^2 (1+6T2+p2T4)2 ( 1 + 6 T^{2} + p^{2} T^{4} )^{2}
17D4×C2D_4\times C_2 1−20T2+358T4−20p2T6+p4T8 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
19D4×C2D_4\times C_2 1−48T2+1118T4−48p2T6+p4T8 1 - 48 T^{2} + 1118 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8}
23D4D_{4} (1−4T+30T2−4pT3+p2T4)2 ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
29D4×C2D_4\times C_2 1−8T2+718T4−8p2T6+p4T8 1 - 8 T^{2} + 718 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}
31D4×C2D_4\times C_2 1−96T2+4046T4−96p2T6+p4T8 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8}
37D4D_{4} (1+8T+70T2+8pT3+p2T4)2 ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
41D4×C2D_4\times C_2 1−116T2+6406T4−116p2T6+p4T8 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8}
43D4×C2D_4\times C_2 1−112T2+6334T4−112p2T6+p4T8 1 - 112 T^{2} + 6334 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8}
47C2C_2 (1−8T+pT2)4 ( 1 - 8 T + p T^{2} )^{4}
53D4×C2D_4\times C_2 1−200T2+15598T4−200p2T6+p4T8 1 - 200 T^{2} + 15598 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1+2T+114T2+2pT3+p2T4)2 ( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
61D4D_{4} (1−8T+118T2−8pT3+p2T4)2 ( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−160T2+13758T4−160p2T6+p4T8 1 - 160 T^{2} + 13758 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1−4T−34T2−4pT3+p2T4)2 ( 1 - 4 T - 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
73C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
79D4×C2D_4\times C_2 1−128T2+7758T4−128p2T6+p4T8 1 - 128 T^{2} + 7758 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8}
83D4D_{4} (1+10T+186T2+10pT3+p2T4)2 ( 1 + 10 T + 186 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}
89C22C_2^2 (1−162T2+p2T4)2 ( 1 - 162 T^{2} + p^{2} T^{4} )^{2}
97D4D_{4} (1+8T+190T2+8pT3+p2T4)2 ( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} 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

−8.218054529458063796990288542526, −8.095712444197151852803827793526, −7.81154220456290406141470382148, −7.31788822670226719981440106942, −7.31167834096556364237367427665, −7.04115419664090146379155592869, −6.87775482818256847204336463769, −6.87608787923098798869876049769, −5.92109346824573554556271882355, −5.90825067597875359038969952359, −5.58033037672935558922287635504, −5.30617010774444546975566291979, −5.06366884368233726996521205517, −5.01193824464423276237475718191, −4.59214357569285728443681311873, −4.08716589593396748958705805894, −3.98058799524650802644509326142, −3.49651607573513232125593208464, −2.99189521466023003254220494210, −2.81637949391504147846896050728, −2.69981003140452455874462258290, −2.38816110549454468176908182544, −2.04424415332624413801861504578, −1.18538501525557478656219182913, −0.61237961889001587365307450298, 0.61237961889001587365307450298, 1.18538501525557478656219182913, 2.04424415332624413801861504578, 2.38816110549454468176908182544, 2.69981003140452455874462258290, 2.81637949391504147846896050728, 2.99189521466023003254220494210, 3.49651607573513232125593208464, 3.98058799524650802644509326142, 4.08716589593396748958705805894, 4.59214357569285728443681311873, 5.01193824464423276237475718191, 5.06366884368233726996521205517, 5.30617010774444546975566291979, 5.58033037672935558922287635504, 5.90825067597875359038969952359, 5.92109346824573554556271882355, 6.87608787923098798869876049769, 6.87775482818256847204336463769, 7.04115419664090146379155592869, 7.31167834096556364237367427665, 7.31788822670226719981440106942, 7.81154220456290406141470382148, 8.095712444197151852803827793526, 8.218054529458063796990288542526

Graph of the ZZ-function along the critical line