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Properties

Label 8-5408e4-1.1-c1e4-0-2
Degree 88
Conductor 8.554×10148.554\times 10^{14}
Sign 11
Analytic cond. 3.47740×1063.47740\times 10^{6}
Root an. cond. 6.571386.57138
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  + 8·5-s − 2·9-s − 12·17-s + 20·25-s + 12·29-s + 24·37-s + 8·41-s − 16·45-s − 6·49-s + 24·53-s + 12·61-s + 24·73-s − 3·81-s − 96·85-s + 16·89-s + 48·97-s + 36·101-s − 24·109-s − 36·113-s − 14·121-s − 40·125-s + 127-s + 131-s + 137-s + 139-s + 96·145-s + 149-s + ⋯
L(s)  = 1  + 3.57·5-s − 2/3·9-s − 2.91·17-s + 4·25-s + 2.22·29-s + 3.94·37-s + 1.24·41-s − 2.38·45-s − 6/7·49-s + 3.29·53-s + 1.53·61-s + 2.80·73-s − 1/3·81-s − 10.4·85-s + 1.69·89-s + 4.87·97-s + 3.58·101-s − 2.29·109-s − 3.38·113-s − 1.27·121-s − 3.57·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 7.97·145-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=((220⋅138)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((220⋅138)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅1382^{20} \cdot 13^{8}
Sign: 11
Analytic conductor: 3.47740×1063.47740\times 10^{6}
Root analytic conductor: 6.571386.57138
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅138, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{20} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 15.6992907515.69929075
L(12)L(\frac12) ≈\approx 15.6992907515.69929075
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
13 1 1
good3C22≀C2C_2^2 \wr C_2 1+2T2+7T4+2p2T6+p4T8 1 + 2 T^{2} + 7 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8}
5C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
7C22≀C2C_2^2 \wr C_2 1+6T2−T4+6p2T6+p4T8 1 + 6 T^{2} - T^{4} + 6 p^{2} T^{6} + p^{4} T^{8}
11C22≀C2C_2^2 \wr C_2 1+14T2+183T4+14p2T6+p4T8 1 + 14 T^{2} + 183 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8}
17D4D_{4} (1+6T+31T2+6pT3+p2T4)2 ( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
19C22≀C2C_2^2 \wr C_2 1+54T2+1343T4+54p2T6+p4T8 1 + 54 T^{2} + 1343 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8}
23C22≀C2C_2^2 \wr C_2 1+10T2+975T4+10p2T6+p4T8 1 + 10 T^{2} + 975 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8}
29C2C_2 (1−3T+pT2)4 ( 1 - 3 T + p T^{2} )^{4}
31C22≀C2C_2^2 \wr C_2 1+36T2+518T4+36p2T6+p4T8 1 + 36 T^{2} + 518 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8}
37C22C_2^2 (1−12T+107T2−12pT3+p2T4)2 ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
41D4D_{4} (1−4T+59T2−4pT3+p2T4)2 ( 1 - 4 T + 59 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
43C22≀C2C_2^2 \wr C_2 1+82T2+4407T4+82p2T6+p4T8 1 + 82 T^{2} + 4407 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8}
47C22≀C2C_2^2 \wr C_2 1−4T2+2694T4−4p2T6+p4T8 1 - 4 T^{2} + 2694 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}
53D4D_{4} (1−12T+130T2−12pT3+p2T4)2 ( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
59C22≀C2C_2^2 \wr C_2 1−10T2+6015T4−10p2T6+p4T8 1 - 10 T^{2} + 6015 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8}
61C2C_2 (1−3T+pT2)4 ( 1 - 3 T + p T^{2} )^{4}
67C22≀C2C_2^2 \wr C_2 1+246T2+23999T4+246p2T6+p4T8 1 + 246 T^{2} + 23999 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8}
71C22≀C2C_2^2 \wr C_2 1+14T2+1383T4+14p2T6+p4T8 1 + 14 T^{2} + 1383 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8}
73C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
79C22≀C2C_2^2 \wr C_2 1+196T2+20358T4+196p2T6+p4T8 1 + 196 T^{2} + 20358 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8}
83C22≀C2C_2^2 \wr C_2 1+212T2+23286T4+212p2T6+p4T8 1 + 212 T^{2} + 23286 T^{4} + 212 p^{2} T^{6} + p^{4} T^{8}
89D4D_{4} (1−8T+167T2−8pT3+p2T4)2 ( 1 - 8 T + 167 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
97D4D_{4} (1−24T+335T2−24pT3+p2T4)2 ( 1 - 24 T + 335 T^{2} - 24 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

−5.94184664639347053353034240805, −5.61577237962366390032595437572, −5.33956919858318088365796375992, −5.28108712777309370804236854173, −5.12677623494920046051003888956, −4.79775757919956575384287121926, −4.75654032116821498723860616807, −4.37038169905296304757194008329, −4.26664682453302570013059360108, −4.04945311624170794286067242408, −3.90142523435735189921468561418, −3.58056082701344505131959704587, −3.41393134280874500600501082507, −2.77429132083988725189528469093, −2.74533127673538015364014662278, −2.55163948399715557602002178049, −2.48868376445881194554485907161, −2.27227925235580273776365267121, −2.05611678814104365367289862196, −1.85615978087165839844504469436, −1.80155011021054350362900125685, −1.21248115921947163502407719856, −0.893703462753866922575439030569, −0.70393521010110959464768181263, −0.46826868496538690293396876483, 0.46826868496538690293396876483, 0.70393521010110959464768181263, 0.893703462753866922575439030569, 1.21248115921947163502407719856, 1.80155011021054350362900125685, 1.85615978087165839844504469436, 2.05611678814104365367289862196, 2.27227925235580273776365267121, 2.48868376445881194554485907161, 2.55163948399715557602002178049, 2.74533127673538015364014662278, 2.77429132083988725189528469093, 3.41393134280874500600501082507, 3.58056082701344505131959704587, 3.90142523435735189921468561418, 4.04945311624170794286067242408, 4.26664682453302570013059360108, 4.37038169905296304757194008329, 4.75654032116821498723860616807, 4.79775757919956575384287121926, 5.12677623494920046051003888956, 5.28108712777309370804236854173, 5.33956919858318088365796375992, 5.61577237962366390032595437572, 5.94184664639347053353034240805

Graph of the ZZ-function along the critical line