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Properties

Label 4-640e2-1.1-c1e2-0-39
Degree 44
Conductor 409600409600
Sign 11
Analytic cond. 26.116426.1164
Root an. cond. 2.260622.26062
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 − 2·5-s + 6·9-s − 4·13-s + 8·15-s − 25-s + 4·27-s − 20·37-s + 16·39-s − 12·41-s − 12·43-s − 12·45-s + 2·49-s + 12·53-s + 8·65-s − 20·67-s + 24·71-s + 4·75-s − 16·79-s − 37·81-s + 12·83-s − 4·89-s + 4·107-s + 80·111-s − 24·117-s − 18·121-s + 48·123-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.894·5-s + 2·9-s − 1.10·13-s + 2.06·15-s − 1/5·25-s + 0.769·27-s − 3.28·37-s + 2.56·39-s − 1.87·41-s − 1.82·43-s − 1.78·45-s + 2/7·49-s + 1.64·53-s + 0.992·65-s − 2.44·67-s + 2.84·71-s + 0.461·75-s − 1.80·79-s − 4.11·81-s + 1.31·83-s − 0.423·89-s + 0.386·107-s + 7.59·111-s − 2.21·117-s − 1.63·121-s + 4.32·123-s + ⋯

Functional equation

Λ(s)=(409600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(409600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 409600409600    =    214⋅522^{14} \cdot 5^{2}
Sign: 11
Analytic conductor: 26.116426.1164
Root analytic conductor: 2.260622.26062
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 409600, ( :1/2,1/2), 1)(4,\ 409600,\ (\ :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+2T+pT2 1 + 2 T + p T^{2}
good3C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
89C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{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

−8.212002059885944684861021878477, −7.50419758061409482324369399006, −6.97464122003923989664965494039, −6.84978263741944475168127235714, −6.33050260025483473718379487161, −5.63664495018734253206101527190, −5.36997692852317792358039338755, −4.84373431553687523132526670248, −4.72524620355206898427648612891, −3.69802392152085758411183661848, −3.36730269661969347724618284285, −2.35167390023705158699342586043, −1.36685449629542271877328763705, 0, 0, 1.36685449629542271877328763705, 2.35167390023705158699342586043, 3.36730269661969347724618284285, 3.69802392152085758411183661848, 4.72524620355206898427648612891, 4.84373431553687523132526670248, 5.36997692852317792358039338755, 5.63664495018734253206101527190, 6.33050260025483473718379487161, 6.84978263741944475168127235714, 6.97464122003923989664965494039, 7.50419758061409482324369399006, 8.212002059885944684861021878477

Graph of the ZZ-function along the critical line