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Properties

Label 4-3960e2-1.1-c0e2-0-3
Degree 44
Conductor 1568160015681600
Sign 11
Analytic cond. 3.905753.90575
Root an. cond. 1.405801.40580
Motivic weight 00
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  + 2·2-s + 3·4-s + 4·8-s + 5·16-s + 4·17-s − 25-s − 4·31-s + 6·32-s + 8·34-s − 2·49-s − 2·50-s − 8·62-s + 7·64-s + 12·68-s − 4·98-s − 3·100-s − 121-s − 12·124-s + 127-s + 8·128-s + 131-s + 16·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 5·16-s + 4·17-s − 25-s − 4·31-s + 6·32-s + 8·34-s − 2·49-s − 2·50-s − 8·62-s + 7·64-s + 12·68-s − 4·98-s − 3·100-s − 121-s − 12·124-s + 127-s + 8·128-s + 131-s + 16·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯

Functional equation

Λ(s)=(15681600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(15681600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1568160015681600    =    26⋅34⋅52⋅1122^{6} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 3.905753.90575
Root analytic conductor: 1.405801.40580
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 15681600, ( :0,0), 1)(4,\ 15681600,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 7.0961142387.096114238
L(12)L(\frac12) ≈\approx 7.0961142387.096114238
L(1)L(1) 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)
bad2C1C_1 (1−T)2 ( 1 - T )^{2}
3 1 1
5C2C_2 1+T2 1 + T^{2}
11C2C_2 1+T2 1 + T^{2}
good7C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
13C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
17C1C_1 (1−T)4 ( 1 - T )^{4}
19C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
23C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
29C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
31C1C_1 (1+T)4 ( 1 + T )^{4}
37C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
41C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
43C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
47C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
53C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
59C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
61C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
67C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
71C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
73C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
79C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
83C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
89C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
97C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + 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

−8.610129517064521063871200790209, −8.231853752708922034883465134851, −7.73457761912609279684093296340, −7.63124012438696137060726654141, −7.21266372005966326085810159255, −7.16929713735343097466295311344, −6.32173419051543895789902153128, −6.02575989141623421100365172667, −5.79965596381305496694007162054, −5.38116083081143528284756870931, −5.09486353341852729646412979775, −4.96518584966744755803389524303, −4.05991564912919533687928125093, −3.72461624341634139208034025797, −3.55230901344726836392846336629, −3.23362658755031601378844349088, −2.74015688377842915809085811906, −2.04403492347991883044002320281, −1.55272786336730007740934220132, −1.25772074744669751086525290423, 1.25772074744669751086525290423, 1.55272786336730007740934220132, 2.04403492347991883044002320281, 2.74015688377842915809085811906, 3.23362658755031601378844349088, 3.55230901344726836392846336629, 3.72461624341634139208034025797, 4.05991564912919533687928125093, 4.96518584966744755803389524303, 5.09486353341852729646412979775, 5.38116083081143528284756870931, 5.79965596381305496694007162054, 6.02575989141623421100365172667, 6.32173419051543895789902153128, 7.16929713735343097466295311344, 7.21266372005966326085810159255, 7.63124012438696137060726654141, 7.73457761912609279684093296340, 8.231853752708922034883465134851, 8.610129517064521063871200790209

Graph of the ZZ-function along the critical line