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Properties

Label 4-1280e2-1.1-c3e2-0-15
Degree 44
Conductor 16384001638400
Sign 11
Analytic cond. 5703.635703.63
Root an. cond. 8.690368.69036
Motivic weight 33
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  + 12·7-s + 50·9-s − 60·17-s + 356·23-s − 25·25-s − 40·31-s + 500·41-s − 428·47-s − 578·49-s + 600·63-s + 200·71-s + 460·73-s + 2.64e3·79-s + 1.77e3·81-s − 1.74e3·89-s − 620·97-s + 2.80e3·103-s − 3.02e3·113-s − 720·119-s − 938·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 3.00e3·153-s + ⋯
L(s)  = 1  + 0.647·7-s + 1.85·9-s − 0.856·17-s + 3.22·23-s − 1/5·25-s − 0.231·31-s + 1.90·41-s − 1.32·47-s − 1.68·49-s + 1.19·63-s + 0.334·71-s + 0.737·73-s + 3.75·79-s + 2.42·81-s − 2.08·89-s − 0.648·97-s + 2.68·103-s − 2.51·113-s − 0.554·119-s − 0.704·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 1.58·153-s + ⋯

Functional equation

Λ(s)=(1638400s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1638400s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 16384001638400    =    216⋅522^{16} \cdot 5^{2}
Sign: 11
Analytic conductor: 5703.635703.63
Root analytic conductor: 8.690368.69036
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1638400, ( :3/2,3/2), 1)(4,\ 1638400,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 4.6554326114.655432611
L(12)L(\frac12) ≈\approx 4.6554326114.655432611
L(52)L(\frac{5}{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
5C2C_2 1+p2T2 1 + p^{2} T^{2}
good3C22C_2^2 1−50T2+p6T4 1 - 50 T^{2} + p^{6} T^{4}
7C2C_2 (1−6T+p3T2)2 ( 1 - 6 T + p^{3} T^{2} )^{2}
11C22C_2^2 1+938T2+p6T4 1 + 938 T^{2} + p^{6} T^{4}
13C22C_2^2 1−1894T2+p6T4 1 - 1894 T^{2} + p^{6} T^{4}
17C2C_2 (1+30T+p3T2)2 ( 1 + 30 T + p^{3} T^{2} )^{2}
19C22C_2^2 1−12118T2+p6T4 1 - 12118 T^{2} + p^{6} T^{4}
23C2C_2 (1−178T+p3T2)2 ( 1 - 178 T + p^{3} T^{2} )^{2}
29C22C_2^2 1−21222T2+p6T4 1 - 21222 T^{2} + p^{6} T^{4}
31C2C_2 (1+20T+p3T2)2 ( 1 + 20 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−101206T2+p6T4 1 - 101206 T^{2} + p^{6} T^{4}
41C2C_2 (1−250T+p3T2)2 ( 1 - 250 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−138850T2+p6T4 1 - 138850 T^{2} + p^{6} T^{4}
47C2C_2 (1+214T+p3T2)2 ( 1 + 214 T + p^{3} T^{2} )^{2}
53C22C_2^2 1−57654T2+p6T4 1 - 57654 T^{2} + p^{6} T^{4}
59C22C_2^2 1+229242T2+p6T4 1 + 229242 T^{2} + p^{6} T^{4}
61C22C_2^2 1−391462T2+p6T4 1 - 391462 T^{2} + p^{6} T^{4}
67C22C_2^2 1−2450T2+p6T4 1 - 2450 T^{2} + p^{6} T^{4}
71C2C_2 (1−100T+p3T2)2 ( 1 - 100 T + p^{3} T^{2} )^{2}
73C2C_2 (1−230T+p3T2)2 ( 1 - 230 T + p^{3} T^{2} )^{2}
79C2C_2 (1−1320T+p3T2)2 ( 1 - 1320 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−179250T2+p6T4 1 - 179250 T^{2} + p^{6} T^{4}
89C2C_2 (1+874T+p3T2)2 ( 1 + 874 T + p^{3} T^{2} )^{2}
97C2C_2 (1+310T+p3T2)2 ( 1 + 310 T + p^{3} 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

−9.488545604752342363081879971220, −9.131089968054584979756754053159, −8.874187213699615644288686830090, −8.129790602217912885143406274182, −7.83814030334685579067299963198, −7.53436504593675575672507494707, −6.86357055416189943977394512739, −6.78146669340449770858592991616, −6.43785196334441514812062745085, −5.68051938983478884997798942980, −4.99512330486873872905173745811, −4.79012015147302774856233374044, −4.64680339207890780948939755518, −3.72450453715078031695408496850, −3.63602816428242171539743033320, −2.69988355327150261673269357363, −2.31716885589250679000601961940, −1.41265709925717057253174356607, −1.29823691443167736644770991193, −0.53820478406691833161297783093, 0.53820478406691833161297783093, 1.29823691443167736644770991193, 1.41265709925717057253174356607, 2.31716885589250679000601961940, 2.69988355327150261673269357363, 3.63602816428242171539743033320, 3.72450453715078031695408496850, 4.64680339207890780948939755518, 4.79012015147302774856233374044, 4.99512330486873872905173745811, 5.68051938983478884997798942980, 6.43785196334441514812062745085, 6.78146669340449770858592991616, 6.86357055416189943977394512739, 7.53436504593675575672507494707, 7.83814030334685579067299963198, 8.129790602217912885143406274182, 8.874187213699615644288686830090, 9.131089968054584979756754053159, 9.488545604752342363081879971220

Graph of the ZZ-function along the critical line