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Properties

Label 4-384e2-1.1-c1e2-0-20
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·5-s + 2·7-s + 9-s − 2·11-s + 4·13-s − 8·15-s + 2·17-s + 2·19-s − 4·21-s + 4·23-s + 2·25-s + 4·27-s + 6·31-s + 4·33-s + 8·35-s + 8·37-s − 8·39-s + 6·41-s − 18·43-s + 4·45-s + 16·47-s − 2·49-s − 4·51-s − 8·55-s − 4·57-s − 22·59-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 2.06·15-s + 0.485·17-s + 0.458·19-s − 0.872·21-s + 0.834·23-s + 2/5·25-s + 0.769·27-s + 1.07·31-s + 0.696·33-s + 1.35·35-s + 1.31·37-s − 1.28·39-s + 0.937·41-s − 2.74·43-s + 0.596·45-s + 2.33·47-s − 2/7·49-s − 0.560·51-s − 1.07·55-s − 0.529·57-s − 2.86·59-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 1.7485138611.748513861
L(12)L(\frac12) ≈\approx 1.7485138611.748513861
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×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2×\timesC2C_2 (1−2T+pT2)(1+4T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 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×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
37C2C_2×\timesC2C_2 (1−10T+pT2)(1+2T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−12T+pT2)(1+6T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2×\timesC2C_2 (1+6T+pT2)(1+12T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2×\timesC2C_2 (1+8T+pT2)(1+14T+pT2) ( 1 + 8 T + p T^{2} )( 1 + 14 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−2T+pT2)(1+10T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−8T+pT2)(1+10T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )
71C2C_2×\timesC2C_2 (1+pT2)(1+12T+pT2) ( 1 + p T^{2} )( 1 + 12 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−14T+pT2)(1−2T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−14T+pT2)(1−8T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−6T+pT2)(1+12T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2×\timesC2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 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

−13.6777112753, −13.3874273048, −12.9029585763, −12.2932744177, −12.0337535527, −11.2885358311, −11.2403489512, −10.5726584783, −10.3699232804, −9.87920052713, −9.21593970226, −9.13514551558, −8.16659881546, −7.98420780715, −7.29562794842, −6.39871498619, −6.33590788591, −5.60222363254, −5.59100833138, −4.83385679588, −4.43324404972, −3.30982302623, −2.63774465597, −1.73703685485, −1.06529864489, 1.06529864489, 1.73703685485, 2.63774465597, 3.30982302623, 4.43324404972, 4.83385679588, 5.59100833138, 5.60222363254, 6.33590788591, 6.39871498619, 7.29562794842, 7.98420780715, 8.16659881546, 9.13514551558, 9.21593970226, 9.87920052713, 10.3699232804, 10.5726584783, 11.2403489512, 11.2885358311, 12.0337535527, 12.2932744177, 12.9029585763, 13.3874273048, 13.6777112753

Graph of the ZZ-function along the critical line