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Properties

Label 4-384e2-1.1-c1e2-0-43
Degree 44
Conductor 147456147456
Sign −1-1
Analytic cond. 9.401929.40192
Root an. cond. 1.751071.75107
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 4·17-s − 6·25-s − 12·41-s − 14·49-s − 20·73-s + 81-s + 12·89-s − 4·97-s + 36·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 1/3·9-s − 0.970·17-s − 6/5·25-s − 1.87·41-s − 2·49-s − 2.34·73-s + 1/9·81-s + 1.27·89-s − 0.406·97-s + 3.38·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-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: −1-1
Analytic conductor: 9.401929.40192
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
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+T2 1 + 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+pT2)2 ( 1 + p T^{2} )^{2}
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 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 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
59C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{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

−8.921254360940967854582257245528, −8.666819353027446340954917061212, −8.113379036905717657653074471412, −7.65564630182415559463917551272, −7.06735387836962857768582860218, −6.58560781686114559363252892690, −6.06492305002903915597915546629, −5.61558786042195542575202104887, −4.81542349107446102730772195937, −4.54676209733795639967608998188, −3.65399134625069874060710032552, −3.20107273236009553461915677614, −2.29353120572318785524143704109, −1.61158418780440794414710143733, 0, 1.61158418780440794414710143733, 2.29353120572318785524143704109, 3.20107273236009553461915677614, 3.65399134625069874060710032552, 4.54676209733795639967608998188, 4.81542349107446102730772195937, 5.61558786042195542575202104887, 6.06492305002903915597915546629, 6.58560781686114559363252892690, 7.06735387836962857768582860218, 7.65564630182415559463917551272, 8.113379036905717657653074471412, 8.666819353027446340954917061212, 8.921254360940967854582257245528

Graph of the ZZ-function along the critical line