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Properties

Label 4-1280e2-1.1-c3e2-0-28
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 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·5-s − 20·9-s − 20·13-s − 84·17-s + 75·25-s + 24·29-s − 364·37-s − 16·41-s − 200·45-s − 652·49-s − 692·53-s + 1.17e3·61-s − 200·65-s − 116·73-s − 329·81-s − 840·85-s + 876·89-s − 2.95e3·97-s − 1.02e3·101-s + 1.50e3·109-s − 3.47e3·113-s + 400·117-s − 2.11e3·121-s + 500·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.740·9-s − 0.426·13-s − 1.19·17-s + 3/5·25-s + 0.153·29-s − 1.61·37-s − 0.0609·41-s − 0.662·45-s − 1.90·49-s − 1.79·53-s + 2.45·61-s − 0.381·65-s − 0.185·73-s − 0.451·81-s − 1.07·85-s + 1.04·89-s − 3.09·97-s − 1.00·101-s + 1.31·109-s − 2.89·113-s + 0.316·117-s − 1.59·121-s + 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-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: 22
Selberg data: (4, 1638400, ( :3/2,3/2), 1)(4,\ 1638400,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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
5C1C_1 (1−pT)2 ( 1 - p T )^{2}
good3C22C_2^2 1+20T2+p6T4 1 + 20 T^{2} + p^{6} T^{4}
7C22C_2^2 1+652T2+p6T4 1 + 652 T^{2} + p^{6} T^{4}
11C22C_2^2 1+2118T2+p6T4 1 + 2118 T^{2} + p^{6} T^{4}
13C2C_2 (1+10T+p3T2)2 ( 1 + 10 T + p^{3} T^{2} )^{2}
17C2C_2 (1+42T+p3T2)2 ( 1 + 42 T + p^{3} T^{2} )^{2}
19C22C_2^2 1+12494T2+p6T4 1 + 12494 T^{2} + p^{6} T^{4}
23C22C_2^2 1+21580T2+p6T4 1 + 21580 T^{2} + p^{6} T^{4}
29C2C_2 (1−12T+p3T2)2 ( 1 - 12 T + p^{3} T^{2} )^{2}
31C22C_2^2 1+43126T2+p6T4 1 + 43126 T^{2} + p^{6} T^{4}
37C2C_2 (1+182T+p3T2)2 ( 1 + 182 T + p^{3} T^{2} )^{2}
41C2C_2 (1+8T+p3T2)2 ( 1 + 8 T + p^{3} T^{2} )^{2}
43C22C_2^2 1+48548T2+p6T4 1 + 48548 T^{2} + p^{6} T^{4}
47C22C_2^2 1+165996T2+p6T4 1 + 165996 T^{2} + p^{6} T^{4}
53C2C_2 (1+346T+p3T2)2 ( 1 + 346 T + p^{3} T^{2} )^{2}
59C22C_2^2 1+159294T2+p6T4 1 + 159294 T^{2} + p^{6} T^{4}
61C2C_2 (1−586T+p3T2)2 ( 1 - 586 T + p^{3} T^{2} )^{2}
67C22C_2^2 1+18052T2+p6T4 1 + 18052 T^{2} + p^{6} T^{4}
71C22C_2^2 1+666726T2+p6T4 1 + 666726 T^{2} + p^{6} T^{4}
73C2C_2 (1+58T+p3T2)2 ( 1 + 58 T + p^{3} T^{2} )^{2}
79C22C_2^2 1−542018T2+p6T4 1 - 542018 T^{2} + p^{6} T^{4}
83C22C_2^2 1+1086420T2+p6T4 1 + 1086420 T^{2} + p^{6} T^{4}
89C2C_2 (1−438T+p3T2)2 ( 1 - 438 T + p^{3} T^{2} )^{2}
97C2C_2 (1+1478T+p3T2)2 ( 1 + 1478 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

−8.990605580361402488415155311932, −8.831143831680356583622079251309, −8.219576812642405089708596251076, −8.115963879653458301978925709490, −7.40823418545090487208834159434, −6.88822619514463943346314866094, −6.47570820356023366291754443605, −6.44262515441773771627314368472, −5.52063210306383044297655800818, −5.46689028324171985792874291383, −4.87829418835866691937258525964, −4.51835617371829475861424651825, −3.80023968503339439830755981082, −3.34126763632260012801337593641, −2.65606522802566493242444111444, −2.39098853092476686762939464930, −1.70046342961834452284189357302, −1.22392467846219956374302175907, 0, 0, 1.22392467846219956374302175907, 1.70046342961834452284189357302, 2.39098853092476686762939464930, 2.65606522802566493242444111444, 3.34126763632260012801337593641, 3.80023968503339439830755981082, 4.51835617371829475861424651825, 4.87829418835866691937258525964, 5.46689028324171985792874291383, 5.52063210306383044297655800818, 6.44262515441773771627314368472, 6.47570820356023366291754443605, 6.88822619514463943346314866094, 7.40823418545090487208834159434, 8.115963879653458301978925709490, 8.219576812642405089708596251076, 8.831143831680356583622079251309, 8.990605580361402488415155311932

Graph of the ZZ-function along the critical line