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Properties

Label 8-416e4-1.1-c1e4-0-2
Degree 88
Conductor 2994837913629948379136
Sign 11
Analytic cond. 121.753121.753
Root an. cond. 1.822571.82257
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 − 6·7-s + 5·9-s + 2·11-s − 4·13-s + 6·17-s − 6·19-s − 12·21-s + 6·23-s − 4·25-s + 10·27-s − 6·29-s + 4·33-s − 2·37-s − 8·39-s + 6·41-s + 6·43-s + 24·47-s + 21·49-s + 12·51-s − 12·57-s − 6·59-s − 14·61-s − 30·63-s − 18·67-s + 12·69-s + 6·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 2.26·7-s + 5/3·9-s + 0.603·11-s − 1.10·13-s + 1.45·17-s − 1.37·19-s − 2.61·21-s + 1.25·23-s − 4/5·25-s + 1.92·27-s − 1.11·29-s + 0.696·33-s − 0.328·37-s − 1.28·39-s + 0.937·41-s + 0.914·43-s + 3.50·47-s + 3·49-s + 1.68·51-s − 1.58·57-s − 0.781·59-s − 1.79·61-s − 3.77·63-s − 2.19·67-s + 1.44·69-s + 0.712·71-s + ⋯

Functional equation

Λ(s)=((220⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((220⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅1342^{20} \cdot 13^{4}
Sign: 11
Analytic conductor: 121.753121.753
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{20} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.2994021272.299402127
L(12)L(\frac12) ≈\approx 2.2994021272.299402127
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
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
good3D4×C2D_4\times C_2 1−2T−T2+2T3+4T4+2pT5−p2T6−2p3T7+p4T8 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
5C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
7D4×C2D_4\times C_2 1+6T+15T2+6pT3+20pT4+6p2T5+15p2T6+6p3T7+p4T8 1 + 6 T + 15 T^{2} + 6 p T^{3} + 20 p T^{4} + 6 p^{2} T^{5} + 15 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1−2T−T2+34T3−140T4+34pT5−p2T6−2p3T7+p4T8 1 - 2 T - T^{2} + 34 T^{3} - 140 T^{4} + 34 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
17D4×C2D_4\times C_2 1−6T+T2−6T3+324T4−6pT5+p2T6−6p3T7+p4T8 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 6 p T^{5} + p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
19D4×C2D_4\times C_2 1+6T+7T2−54T3−204T4−54pT5+7p2T6+6p3T7+p4T8 1 + 6 T + 7 T^{2} - 54 T^{3} - 204 T^{4} - 54 p T^{5} + 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1−6T−T2+54T3+12T4+54pT5−p2T6−6p3T7+p4T8 1 - 6 T - T^{2} + 54 T^{3} + 12 T^{4} + 54 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1+6T+T2−138T3−660T4−138pT5+p2T6+6p3T7+p4T8 1 + 6 T + T^{2} - 138 T^{3} - 660 T^{4} - 138 p T^{5} + p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
31C22C_2^2 (1+30T2+p2T4)2 ( 1 + 30 T^{2} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1+2T+T2−142T3−1508T4−142pT5+p2T6+2p3T7+p4T8 1 + 2 T + T^{2} - 142 T^{3} - 1508 T^{4} - 142 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
41D4×C2D_4\times C_2 1−6T−47T2−6T3+3732T4−6pT5−47p2T6−6p3T7+p4T8 1 - 6 T - 47 T^{2} - 6 T^{3} + 3732 T^{4} - 6 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
43D4×C2D_4\times C_2 1−6T−9T2+246T3−1372T4+246pT5−9p2T6−6p3T7+p4T8 1 - 6 T - 9 T^{2} + 246 T^{3} - 1372 T^{4} + 246 p T^{5} - 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
47C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
53C22C_2^2 (1+98T2+p2T4)2 ( 1 + 98 T^{2} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1+6T−73T2−54T3+6276T4−54pT5−73p2T6+6p3T7+p4T8 1 + 6 T - 73 T^{2} - 54 T^{3} + 6276 T^{4} - 54 p T^{5} - 73 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
61C22C_2^2 (1+7T−12T2+7pT3+p2T4)2 ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1+18T+127T2+1134T3+12612T4+1134pT5+127p2T6+18p3T7+p4T8 1 + 18 T + 127 T^{2} + 1134 T^{3} + 12612 T^{4} + 1134 p T^{5} + 127 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}
71D4×C2D_4\times C_2 1−6T−97T2+54T3+10092T4+54pT5−97p2T6−6p3T7+p4T8 1 - 6 T - 97 T^{2} + 54 T^{3} + 10092 T^{4} + 54 p T^{5} - 97 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
73D4D_{4} (1−8T+90T2−8pT3+p2T4)2 ( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
79C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
83C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
89D4×C2D_4\times C_2 1−18T+97T2−882T3+14772T4−882pT5+97p2T6−18p3T7+p4T8 1 - 18 T + 97 T^{2} - 882 T^{3} + 14772 T^{4} - 882 p T^{5} + 97 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}
97C22C_2^2 (1−9T−16T2−9pT3+p2T4)2 ( 1 - 9 T - 16 T^{2} - 9 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

−8.122432780260744564128465704578, −7.903609832611024544152735507094, −7.36033793133985790473398221516, −7.35168825284763161452944805537, −7.33462485768820063904887853516, −7.11120684372679080316926938719, −6.46960408175150434653360802389, −6.39630901918497497806049190205, −6.38828120616388474176376717509, −5.81696401799114879105412283180, −5.67336017977472620843637260124, −5.45303269820738617530664008474, −4.92692760236152258859330306472, −4.67384186769032496677940069073, −4.26256629918312557751062620821, −4.04840041830719252535537556835, −3.71599974571601652190490860191, −3.70892945981940961946842011871, −3.07031960212961596442441316760, −2.96603595367112199857029046969, −2.47081857615985642349475466340, −2.44918310669262376538627162911, −1.76323639653083310184307739574, −1.25426686852587588783194986356, −0.54647633518433370662244596702, 0.54647633518433370662244596702, 1.25426686852587588783194986356, 1.76323639653083310184307739574, 2.44918310669262376538627162911, 2.47081857615985642349475466340, 2.96603595367112199857029046969, 3.07031960212961596442441316760, 3.70892945981940961946842011871, 3.71599974571601652190490860191, 4.04840041830719252535537556835, 4.26256629918312557751062620821, 4.67384186769032496677940069073, 4.92692760236152258859330306472, 5.45303269820738617530664008474, 5.67336017977472620843637260124, 5.81696401799114879105412283180, 6.38828120616388474176376717509, 6.39630901918497497806049190205, 6.46960408175150434653360802389, 7.11120684372679080316926938719, 7.33462485768820063904887853516, 7.35168825284763161452944805537, 7.36033793133985790473398221516, 7.903609832611024544152735507094, 8.122432780260744564128465704578

Graph of the ZZ-function along the critical line