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Properties

Label 4-160e2-1.1-c1e2-0-22
Degree 44
Conductor 2560025600
Sign 11
Analytic cond. 1.632271.63227
Root an. cond. 1.130311.13031
Motivic weight 11
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  − 4·3-s − 4·5-s − 4·7-s + 8·9-s − 2·13-s + 16·15-s − 10·17-s − 8·19-s + 16·21-s − 4·23-s + 11·25-s − 12·27-s + 16·35-s + 2·37-s + 8·39-s + 12·43-s − 32·45-s + 4·47-s + 8·49-s + 40·51-s − 14·53-s + 32·57-s − 8·59-s − 8·61-s − 32·63-s + 8·65-s + 20·67-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.78·5-s − 1.51·7-s + 8/3·9-s − 0.554·13-s + 4.13·15-s − 2.42·17-s − 1.83·19-s + 3.49·21-s − 0.834·23-s + 11/5·25-s − 2.30·27-s + 2.70·35-s + 0.328·37-s + 1.28·39-s + 1.82·43-s − 4.77·45-s + 0.583·47-s + 8/7·49-s + 5.60·51-s − 1.92·53-s + 4.23·57-s − 1.04·59-s − 1.02·61-s − 4.03·63-s + 0.992·65-s + 2.44·67-s + ⋯

Functional equation

Λ(s)=(25600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(25600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2560025600    =    210⋅522^{10} \cdot 5^{2}
Sign: 11
Analytic conductor: 1.632271.63227
Root analytic conductor: 1.130311.13031
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 25600, ( :1/2,1/2), 1)(4,\ 25600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
good3C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
7C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+6T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
29C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
37C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C22C_2^2 1−12T+72T2−12pT3+p2T4 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+14T+98T2+14pT3+p2T4 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4}
59C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
61C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
67C22C_2^2 1−20T+200T2−20pT3+p2T4 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
73C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
79C2C_2 (1+16T+pT2)2 ( 1 + 16 T + p T^{2} )^{2}
83C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
89C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
97C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
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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

−12.39120829607980716833016442920, −12.24639074133331883143606070816, −11.53149773793812008891315253886, −11.21213632369939348194190299961, −10.72216362126657541072761406516, −10.65330872072270715417064095239, −9.694235667765133118993808626927, −9.155775512676237659889010969563, −8.493258516651058848658419487236, −7.79991666445786904202477652183, −6.98204015859890003664557088210, −6.76496165610320624134231613644, −6.17238974846732228512913187291, −5.82427128957150840200781620625, −4.62121038175294800783863441427, −4.44760348718601702809622473482, −3.81985278896428350058783924405, −2.55316889778586010739924050045, 0, 0, 2.55316889778586010739924050045, 3.81985278896428350058783924405, 4.44760348718601702809622473482, 4.62121038175294800783863441427, 5.82427128957150840200781620625, 6.17238974846732228512913187291, 6.76496165610320624134231613644, 6.98204015859890003664557088210, 7.79991666445786904202477652183, 8.493258516651058848658419487236, 9.155775512676237659889010969563, 9.694235667765133118993808626927, 10.65330872072270715417064095239, 10.72216362126657541072761406516, 11.21213632369939348194190299961, 11.53149773793812008891315253886, 12.24639074133331883143606070816, 12.39120829607980716833016442920

Graph of the ZZ-function along the critical line