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Properties

Label 4-384e2-1.1-c1e2-0-51
Degree 44
Conductor 147456147456
Sign 11
Analytic cond. 9.401929.40192
Root an. cond. 1.751071.75107
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  − 2·3-s − 8·7-s + 9-s − 4·13-s − 4·19-s + 16·21-s − 6·25-s + 4·27-s − 20·37-s + 8·39-s − 12·43-s + 34·49-s + 8·57-s − 4·61-s − 8·63-s − 20·67-s + 28·73-s + 12·75-s − 16·79-s − 11·81-s + 32·91-s − 4·97-s − 8·103-s + 12·109-s + 40·111-s − 4·117-s − 18·121-s + ⋯
L(s)  = 1  − 1.15·3-s − 3.02·7-s + 1/3·9-s − 1.10·13-s − 0.917·19-s + 3.49·21-s − 6/5·25-s + 0.769·27-s − 3.28·37-s + 1.28·39-s − 1.82·43-s + 34/7·49-s + 1.05·57-s − 0.512·61-s − 1.00·63-s − 2.44·67-s + 3.27·73-s + 1.38·75-s − 1.80·79-s − 1.22·81-s + 3.35·91-s − 0.406·97-s − 0.788·103-s + 1.14·109-s + 3.79·111-s − 0.369·117-s − 1.63·121-s + ⋯

Functional equation

Λ(s)=(147456s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(147456s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 147456147456    =    214⋅322^{14} \cdot 3^{2}
Sign: 11
Analytic conductor: 9.401929.40192
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 147456, ( :1/2,1/2), 1)(4,\ 147456,\ (\ :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
3C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{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)2 ( 1 + 2 T + p T^{2} )^{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)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{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)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{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)2 ( 1 + 2 T + p T^{2} )^{2}
67C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
71C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
97C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p 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

−8.759776534720553396196404795081, −8.635939205168057076823393046651, −7.62957697295837367057177448880, −6.97464122003923989664965494039, −6.80509880453900742773724680138, −6.33050260025483473718379487161, −5.94298026933117547539540451748, −5.37965797861855942151102168788, −4.84373431553687523132526670248, −3.98566652809305192679300882554, −3.36730269661969347724618284285, −2.97266024838274409285785943283, −1.98277685475006180754048515240, 0, 0, 1.98277685475006180754048515240, 2.97266024838274409285785943283, 3.36730269661969347724618284285, 3.98566652809305192679300882554, 4.84373431553687523132526670248, 5.37965797861855942151102168788, 5.94298026933117547539540451748, 6.33050260025483473718379487161, 6.80509880453900742773724680138, 6.97464122003923989664965494039, 7.62957697295837367057177448880, 8.635939205168057076823393046651, 8.759776534720553396196404795081

Graph of the ZZ-function along the critical line