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Properties

Label 8-768e4-1.1-c1e4-0-5
Degree 88
Conductor 347892350976347892350976
Sign 11
Analytic cond. 1414.331414.33
Root an. cond. 2.476392.47639
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 − 4·5-s + 2·9-s + 8·15-s + 12·19-s + 8·23-s − 6·27-s + 28·29-s + 20·43-s − 8·45-s − 32·47-s + 16·49-s − 4·53-s − 24·57-s + 12·67-s − 16·69-s + 8·71-s + 8·73-s + 11·81-s − 56·87-s − 48·95-s − 16·97-s − 20·101-s − 32·115-s + 16·121-s + 20·125-s + 127-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 2/3·9-s + 2.06·15-s + 2.75·19-s + 1.66·23-s − 1.15·27-s + 5.19·29-s + 3.04·43-s − 1.19·45-s − 4.66·47-s + 16/7·49-s − 0.549·53-s − 3.17·57-s + 1.46·67-s − 1.92·69-s + 0.949·71-s + 0.936·73-s + 11/9·81-s − 6.00·87-s − 4.92·95-s − 1.62·97-s − 1.99·101-s − 2.98·115-s + 1.45·121-s + 1.78·125-s + 0.0887·127-s + ⋯

Functional equation

Λ(s)=((232⋅34)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((232⋅34)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \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: 232⋅342^{32} \cdot 3^{4}
Sign: 11
Analytic conductor: 1414.331414.33
Root analytic conductor: 2.476392.47639
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 232⋅34, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.3680559161.368055916
L(12)L(\frac12) ≈\approx 1.3680559161.368055916
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}
good5D4D_{4} (1+2T+6T2+2pT3+p2T4)2 ( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
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}
11C4×C2C_4\times C_2 1−16T2+126T4−16p2T6+p4T8 1 - 16 T^{2} + 126 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}
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}
19D4D_{4} (1−6T+42T2−6pT3+p2T4)2 ( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
23D4D_{4} (1−4T+30T2−4pT3+p2T4)2 ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
29D4D_{4} (1−14T+102T2−14pT3+p2T4)2 ( 1 - 14 T + 102 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}
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}
37D4×C2D_4\times C_2 1−76T2+2902T4−76p2T6+p4T8 1 - 76 T^{2} + 2902 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8}
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}
43D4D_{4} (1−10T+106T2−10pT3+p2T4)2 ( 1 - 10 T + 106 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}
47C2C_2 (1+8T+pT2)4 ( 1 + 8 T + p T^{2} )^{4}
53D4D_{4} (1+2T+102T2+2pT3+p2T4)2 ( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1−224T2+19486T4−224p2T6+p4T8 1 - 224 T^{2} + 19486 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8}
61D4×C2D_4\times C_2 1−172T2+13558T4−172p2T6+p4T8 1 - 172 T^{2} + 13558 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8}
67D4D_{4} (1−6T+98T2−6pT3+p2T4)2 ( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
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}
83D4×C2D_4\times C_2 1−272T2+31774T4−272p2T6+p4T8 1 - 272 T^{2} + 31774 T^{4} - 272 p^{2} T^{6} + p^{4} T^{8}
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

−7.57992738761191657202383396808, −7.01588506882836258599405623548, −6.84775663259973337774910849398, −6.83332850925299462284188648999, −6.71368811642471717302431040962, −6.21752813489160752839231896883, −6.05194532522764385833944496954, −5.66507361989607478536249095400, −5.62772530751221583752710294768, −5.17121339150905459758234223314, −4.91999316105519288251409516356, −4.82030855904975558518792360855, −4.78066862679563805895927793679, −4.13399543037172595332335176138, −4.07998570040077863170079467191, −3.88490084153041906223180962243, −3.37140658726089546864194931169, −3.11232730619813553288894916702, −3.03795318017681575240638062702, −2.62124695530669829008843234732, −2.33752077307676352294556198400, −1.57529871934387436279978745277, −1.02081493097523189981406554888, −0.981546609604123035453254277812, −0.47925865927692612212045167709, 0.47925865927692612212045167709, 0.981546609604123035453254277812, 1.02081493097523189981406554888, 1.57529871934387436279978745277, 2.33752077307676352294556198400, 2.62124695530669829008843234732, 3.03795318017681575240638062702, 3.11232730619813553288894916702, 3.37140658726089546864194931169, 3.88490084153041906223180962243, 4.07998570040077863170079467191, 4.13399543037172595332335176138, 4.78066862679563805895927793679, 4.82030855904975558518792360855, 4.91999316105519288251409516356, 5.17121339150905459758234223314, 5.62772530751221583752710294768, 5.66507361989607478536249095400, 6.05194532522764385833944496954, 6.21752813489160752839231896883, 6.71368811642471717302431040962, 6.83332850925299462284188648999, 6.84775663259973337774910849398, 7.01588506882836258599405623548, 7.57992738761191657202383396808

Graph of the ZZ-function along the critical line