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Properties

Label 8-132e4-1.1-c2e4-0-0
Degree 88
Conductor 303595776303595776
Sign 11
Analytic cond. 167.353167.353
Root an. cond. 1.896501.89650
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  − 4·2-s + 4·4-s + 4·5-s + 16·8-s − 6·9-s − 16·10-s + 20·13-s − 64·16-s − 80·17-s + 24·18-s + 16·20-s − 24·25-s − 80·26-s + 112·29-s + 64·32-s + 320·34-s − 24·36-s − 40·37-s + 64·40-s − 8·41-s − 24·45-s − 8·49-s + 96·50-s + 80·52-s + 52·53-s − 448·58-s − 4·61-s + ⋯
L(s)  = 1  − 2·2-s + 4-s + 4/5·5-s + 2·8-s − 2/3·9-s − 8/5·10-s + 1.53·13-s − 4·16-s − 4.70·17-s + 4/3·18-s + 4/5·20-s − 0.959·25-s − 3.07·26-s + 3.86·29-s + 2·32-s + 9.41·34-s − 2/3·36-s − 1.08·37-s + 8/5·40-s − 0.195·41-s − 0.533·45-s − 0.163·49-s + 1.91·50-s + 1.53·52-s + 0.981·53-s − 7.72·58-s − 0.0655·61-s + ⋯

Functional equation

Λ(s)=((28⋅34⋅114)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((28⋅34⋅114)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28⋅34⋅1142^{8} \cdot 3^{4} \cdot 11^{4}
Sign: 11
Analytic conductor: 167.353167.353
Root analytic conductor: 1.896501.89650
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅34⋅114, ( :1,1,1,1), 1)(8,\ 2^{8} \cdot 3^{4} \cdot 11^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.40449590140.4044959014
L(12)L(\frac12) ≈\approx 0.40449590140.4044959014
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)
bad2C2C_2 (1+pT+p2T2)2 ( 1 + p T + p^{2} T^{2} )^{2}
3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
good5D4D_{4} (1−2T+18T2−2p2T3+p4T4)2 ( 1 - 2 T + 18 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2}
7D4×C2D_4\times C_2 1+8T2+3630T4+8p4T6+p8T8 1 + 8 T^{2} + 3630 T^{4} + 8 p^{4} T^{6} + p^{8} T^{8}
13D4D_{4} (1−10T+330T2−10p2T3+p4T4)2 ( 1 - 10 T + 330 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2}
17D4D_{4} (1+40T+846T2+40p2T3+p4T4)2 ( 1 + 40 T + 846 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2}
19D4×C2D_4\times C_2 1−628T2+340230T4−628p4T6+p8T8 1 - 628 T^{2} + 340230 T^{4} - 628 p^{4} T^{6} + p^{8} T^{8}
23D4×C2D_4\times C_2 1−1192T2+771150T4−1192p4T6+p8T8 1 - 1192 T^{2} + 771150 T^{4} - 1192 p^{4} T^{6} + p^{8} T^{8}
29D4D_{4} (1−56T+2334T2−56p2T3+p4T4)2 ( 1 - 56 T + 2334 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2}
31D4×C2D_4\times C_2 1−292T2+651846T4−292p4T6+p8T8 1 - 292 T^{2} + 651846 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8}
37D4D_{4} (1+20T+2310T2+20p2T3+p4T4)2 ( 1 + 20 T + 2310 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2}
41D4D_{4} (1+4T−1386T2+4p2T3+p4T4)2 ( 1 + 4 T - 1386 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2}
43D4×C2D_4\times C_2 1−3844T2+9315174T4−3844p4T6+p8T8 1 - 3844 T^{2} + 9315174 T^{4} - 3844 p^{4} T^{6} + p^{8} T^{8}
47D4×C2D_4\times C_2 1−6568T2+19677966T4−6568p4T6+p8T8 1 - 6568 T^{2} + 19677966 T^{4} - 6568 p^{4} T^{6} + p^{8} T^{8}
53D4D_{4} (1−26T+5490T2−26p2T3+p4T4)2 ( 1 - 26 T + 5490 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2}
59D4×C2D_4\times C_2 1−6580T2+33519174T4−6580p4T6+p8T8 1 - 6580 T^{2} + 33519174 T^{4} - 6580 p^{4} T^{6} + p^{8} T^{8}
61D4D_{4} (1+2T+4770T2+2p2T3+p4T4)2 ( 1 + 2 T + 4770 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2}
67C22C_2^2 (1−5090T2+p4T4)2 ( 1 - 5090 T^{2} + p^{4} T^{4} )^{2}
71D4×C2D_4\times C_2 1−12616T2+80339214T4−12616p4T6+p8T8 1 - 12616 T^{2} + 80339214 T^{4} - 12616 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1−124T+9750T2−124p2T3+p4T4)2 ( 1 - 124 T + 9750 T^{2} - 124 p^{2} T^{3} + p^{4} T^{4} )^{2}
79D4×C2D_4\times C_2 1−40pT2−3030738T4−40p5T6+p8T8 1 - 40 p T^{2} - 3030738 T^{4} - 40 p^{5} T^{6} + p^{8} T^{8}
83C22C_2^2 (1−7970T2+p4T4)2 ( 1 - 7970 T^{2} + p^{4} T^{4} )^{2}
89D4D_{4} (1+292T+36630T2+292p2T3+p4T4)2 ( 1 + 292 T + 36630 T^{2} + 292 p^{2} T^{3} + p^{4} T^{4} )^{2}
97D4D_{4} (1−304T+40734T2−304p2T3+p4T4)2 ( 1 - 304 T + 40734 T^{2} - 304 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

−9.601982183708599305463853459271, −9.025409014267321491101276828609, −8.784797082128610098993963790980, −8.755007666286781275135792556189, −8.494129318940361482779672913276, −8.431738694084789791469104744428, −8.285230064489809864411120973835, −7.53236642497542807058389878783, −7.36088472163341506724623016989, −6.96595809543905205644989023848, −6.55932500387201033714313396907, −6.48338041316652082212429707920, −6.22175265381183989063741286740, −5.87530289969919848971704589453, −5.20118511433423897695568674561, −4.77466957829398070274686097400, −4.66607232909690556237524073255, −4.38206677650804563255956093389, −3.88416871261637369566472642143, −3.45213666022171266527022024432, −2.57395257737984768566536002446, −2.13480215132067126613102776406, −2.00484079168505329363101560549, −1.09903719916065301487479286413, −0.40816399520615971761058231308, 0.40816399520615971761058231308, 1.09903719916065301487479286413, 2.00484079168505329363101560549, 2.13480215132067126613102776406, 2.57395257737984768566536002446, 3.45213666022171266527022024432, 3.88416871261637369566472642143, 4.38206677650804563255956093389, 4.66607232909690556237524073255, 4.77466957829398070274686097400, 5.20118511433423897695568674561, 5.87530289969919848971704589453, 6.22175265381183989063741286740, 6.48338041316652082212429707920, 6.55932500387201033714313396907, 6.96595809543905205644989023848, 7.36088472163341506724623016989, 7.53236642497542807058389878783, 8.285230064489809864411120973835, 8.431738694084789791469104744428, 8.494129318940361482779672913276, 8.755007666286781275135792556189, 8.784797082128610098993963790980, 9.025409014267321491101276828609, 9.601982183708599305463853459271

Graph of the ZZ-function along the critical line