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Properties

Label 4-2e6-1.1-c4e2-0-0
Degree 44
Conductor 6464
Sign 11
Analytic cond. 0.6838620.683862
Root an. cond. 0.9093730.909373
Motivic weight 44
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  − 2·2-s + 12·3-s − 12·4-s − 24·6-s + 56·8-s − 54·9-s − 52·11-s − 144·12-s + 80·16-s + 452·17-s + 108·18-s + 268·19-s + 104·22-s + 672·24-s + 290·25-s − 2.05e3·27-s − 1.05e3·32-s − 624·33-s − 904·34-s + 648·36-s − 536·38-s + 1.98e3·41-s − 3.76e3·43-s + 624·44-s + 960·48-s + 962·49-s − 580·50-s + ⋯
L(s)  = 1  − 1/2·2-s + 4/3·3-s − 3/4·4-s − 2/3·6-s + 7/8·8-s − 2/3·9-s − 0.429·11-s − 12-s + 5/16·16-s + 1.56·17-s + 1/3·18-s + 0.742·19-s + 0.214·22-s + 7/6·24-s + 0.463·25-s − 2.81·27-s − 1.03·32-s − 0.573·33-s − 0.782·34-s + 1/2·36-s − 0.371·38-s + 1.18·41-s − 2.03·43-s + 0.322·44-s + 5/12·48-s + 0.400·49-s − 0.231·50-s + ⋯

Functional equation

Λ(s)=(64s/2 ΓC(s)2 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
Λ(s)=(64s/2 ΓC(s+2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6464    =    262^{6}
Sign: 11
Analytic conductor: 0.6838620.683862
Root analytic conductor: 0.9093730.909373
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 64, ( :2,2), 1)(4,\ 64,\ (\ :2, 2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.86747212870.8674721287
L(12)L(\frac12) ≈\approx 0.86747212870.8674721287
L(3)L(3) 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+p4T2 1 + p T + p^{4} T^{2}
good3C2C_2 (1−2pT+p4T2)2 ( 1 - 2 p T + p^{4} T^{2} )^{2}
5C22C_2^2 1−58pT2+p8T4 1 - 58 p T^{2} + p^{8} T^{4}
7C22C_2^2 1−962T2+p8T4 1 - 962 T^{2} + p^{8} T^{4}
11C2C_2 (1+26T+p4T2)2 ( 1 + 26 T + p^{4} T^{2} )^{2}
13C22C_2^2 1−56162T2+p8T4 1 - 56162 T^{2} + p^{8} T^{4}
17C2C_2 (1−226T+p4T2)2 ( 1 - 226 T + p^{4} T^{2} )^{2}
19C2C_2 (1−134T+p4T2)2 ( 1 - 134 T + p^{4} T^{2} )^{2}
23C22C_2^2 1−463682T2+p8T4 1 - 463682 T^{2} + p^{8} T^{4}
29C22C_2^2 1−1298402T2+p8T4 1 - 1298402 T^{2} + p^{8} T^{4}
31C22C_2^2 1−311042T2+p8T4 1 - 311042 T^{2} + p^{8} T^{4}
37C22C_2^2 1−629282T2+p8T4 1 - 629282 T^{2} + p^{8} T^{4}
41C2C_2 (1−994T+p4T2)2 ( 1 - 994 T + p^{4} T^{2} )^{2}
43C2C_2 (1+1882T+p4T2)2 ( 1 + 1882 T + p^{4} T^{2} )^{2}
47C22C_2^2 1−5320322T2+p8T4 1 - 5320322 T^{2} + p^{8} T^{4}
53C22C_2^2 1−1257122T2+p8T4 1 - 1257122 T^{2} + p^{8} T^{4}
59C2C_2 (1+5018T+p4T2)2 ( 1 + 5018 T + p^{4} T^{2} )^{2}
61C22C_2^2 1−23382242T2+p8T4 1 - 23382242 T^{2} + p^{8} T^{4}
67C2C_2 (1−8006T+p4T2)2 ( 1 - 8006 T + p^{4} T^{2} )^{2}
71C22C_2^2 1−50512322T2+p8T4 1 - 50512322 T^{2} + p^{8} T^{4}
73C2C_2 (1−386T+p4T2)2 ( 1 - 386 T + p^{4} T^{2} )^{2}
79C22C_2^2 1+43766398T2+p8T4 1 + 43766398 T^{2} + p^{8} T^{4}
83C2C_2 (1+2234T+p4T2)2 ( 1 + 2234 T + p^{4} T^{2} )^{2}
89C2C_2 (1+10046T+p4T2)2 ( 1 + 10046 T + p^{4} T^{2} )^{2}
97C2C_2 (1−8738T+p4T2)2 ( 1 - 8738 T + p^{4} T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−21.41974751600567597501507952751, −20.75851879792913376932067769058, −19.88017054250292570081466802057, −19.75554523375872504894024081820, −18.57549211362932464980201114502, −18.47094474270830168394216425506, −17.11849225917962360839081028424, −16.84746508040984966204166572848, −15.59975009337965161486346141718, −14.54604406493841985013454267836, −14.18644411469039162318032379072, −13.54086326319204164680456829676, −12.49505108077315281617686365600, −11.25256454262619031776370893028, −9.965584594603764605037194581979, −9.223835836731734465687979184813, −8.300859782990254159011153936074, −7.73365253042416185622777892779, −5.42938113619192888783297542980, −3.27029599977862769934910771922, 3.27029599977862769934910771922, 5.42938113619192888783297542980, 7.73365253042416185622777892779, 8.300859782990254159011153936074, 9.223835836731734465687979184813, 9.965584594603764605037194581979, 11.25256454262619031776370893028, 12.49505108077315281617686365600, 13.54086326319204164680456829676, 14.18644411469039162318032379072, 14.54604406493841985013454267836, 15.59975009337965161486346141718, 16.84746508040984966204166572848, 17.11849225917962360839081028424, 18.47094474270830168394216425506, 18.57549211362932464980201114502, 19.75554523375872504894024081820, 19.88017054250292570081466802057, 20.75851879792913376932067769058, 21.41974751600567597501507952751

Graph of the ZZ-function along the critical line