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Properties

Label 4-384e2-1.1-c1e2-0-31
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 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·5-s + 9-s + 8·15-s + 4·19-s + 8·23-s + 2·25-s − 4·27-s − 12·29-s + 12·43-s + 4·45-s − 16·47-s + 2·49-s − 12·53-s + 8·57-s + 20·67-s + 16·69-s + 24·71-s + 28·73-s + 4·75-s − 11·81-s − 24·87-s + 16·95-s − 4·97-s − 12·101-s + 32·115-s − 18·121-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 1/3·9-s + 2.06·15-s + 0.917·19-s + 1.66·23-s + 2/5·25-s − 0.769·27-s − 2.22·29-s + 1.82·43-s + 0.596·45-s − 2.33·47-s + 2/7·49-s − 1.64·53-s + 1.05·57-s + 2.44·67-s + 1.92·69-s + 2.84·71-s + 3.27·73-s + 0.461·75-s − 1.22·81-s − 2.57·87-s + 1.64·95-s − 0.406·97-s − 1.19·101-s + 2.98·115-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: 00
Selberg data: (4, 147456, ( :1/2,1/2), 1)(4,\ 147456,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3257993283.325799328
L(12)L(\frac12) ≈\approx 3.3257993283.325799328
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)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)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{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)2 ( 1 - 4 T + p T^{2} )^{2}
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{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)2 ( 1 + 8 T + p T^{2} )^{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)2 ( 1 - 14 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} )
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

−9.385355573475711088007535446099, −9.174652574334721806401375408825, −8.173183686627627779120693205031, −8.075661729852922503369502159976, −7.44440089656946414681700247943, −6.78571919296342801026858814862, −6.40345159211036072533921924868, −5.64130249043000575312616744005, −5.36530815656846370209456927790, −4.88235750431299832390164653972, −3.62014189552160099547672624510, −3.56276783709589056726776122484, −2.46608091328452001690059553408, −2.20034159198920039710092677140, −1.34910975935301153966920068234, 1.34910975935301153966920068234, 2.20034159198920039710092677140, 2.46608091328452001690059553408, 3.56276783709589056726776122484, 3.62014189552160099547672624510, 4.88235750431299832390164653972, 5.36530815656846370209456927790, 5.64130249043000575312616744005, 6.40345159211036072533921924868, 6.78571919296342801026858814862, 7.44440089656946414681700247943, 8.075661729852922503369502159976, 8.173183686627627779120693205031, 9.174652574334721806401375408825, 9.385355573475711088007535446099

Graph of the ZZ-function along the critical line