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Properties

Label 4-1200e2-1.1-c1e2-0-18
Degree 44
Conductor 14400001440000
Sign 11
Analytic cond. 91.815691.8156
Root an. cond. 3.095483.09548
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s + 6·13-s + 2·17-s − 12·29-s − 10·37-s + 8·41-s + 10·49-s + 18·53-s + 12·61-s − 18·73-s + 81-s + 20·89-s + 6·97-s − 4·101-s + 4·109-s − 14·113-s + 6·117-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1/3·9-s + 1.66·13-s + 0.485·17-s − 2.22·29-s − 1.64·37-s + 1.24·41-s + 10/7·49-s + 2.47·53-s + 1.53·61-s − 2.10·73-s + 1/9·81-s + 2.11·89-s + 0.609·97-s − 0.398·101-s + 0.383·109-s − 1.31·113-s + 0.554·117-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 + 0.161·153-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=(1440000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1440000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14400001440000    =    28⋅32⋅542^{8} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 91.815691.8156
Root analytic conductor: 3.095483.09548
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1440000, ( :1/2,1/2), 1)(4,\ 1440000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3929155042.392915504
L(12)L(\frac12) ≈\approx 2.3929155042.392915504
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
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
5 1 1
good7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
17C2C_2×\timesC2C_2 (1−2T+pT2)(1+pT2) ( 1 - 2 T + p T^{2} )( 1 + p T^{2} )
19C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
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)2 ( 1 + 6 T + p T^{2} )^{2}
31C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
43C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−12T+pT2)(1−6T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} )
59C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−10T+pT2)(1−2T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} )
67C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
71C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1+4T+pT2)(1+14T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} )
79C22C_2^2 1+138T2+p2T4 1 + 138 T^{2} + p^{2} T^{4}
83C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1−14T+pT2)(1−6T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−12T+pT2)(1+6T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 6 T + p 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

−7.83540487041894339635689883732, −7.41821810397637105155494485920, −7.19929019076115046584976214862, −6.66395544733240397854523548625, −6.10837574039395591018955960628, −5.66019988425805891741289162331, −5.50531724902698633695346568170, −4.88685525944060865041314880641, −4.06522363185768058926557610456, −3.83068347409164253221398360600, −3.55761953444439858212278594982, −2.72632634991445829166818625811, −2.07218279554419329920761833926, −1.46256558013774941495503677051, −0.71054314229633382874325179743, 0.71054314229633382874325179743, 1.46256558013774941495503677051, 2.07218279554419329920761833926, 2.72632634991445829166818625811, 3.55761953444439858212278594982, 3.83068347409164253221398360600, 4.06522363185768058926557610456, 4.88685525944060865041314880641, 5.50531724902698633695346568170, 5.66019988425805891741289162331, 6.10837574039395591018955960628, 6.66395544733240397854523548625, 7.19929019076115046584976214862, 7.41821810397637105155494485920, 7.83540487041894339635689883732

Graph of the ZZ-function along the critical line