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Properties

Label 4-30e4-1.1-c1e2-0-7
Degree 44
Conductor 810000810000
Sign 11
Analytic cond. 51.646351.6463
Root an. cond. 2.680772.68077
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  + 14·19-s + 22·31-s + 13·49-s − 2·61-s + 8·79-s − 34·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 3.21·19-s + 3.95·31-s + 13/7·49-s − 0.256·61-s + 0.900·79-s − 3.25·109-s − 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯

Functional equation

Λ(s)=(810000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(810000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 810000810000    =    24⋅34⋅542^{4} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 51.646351.6463
Root analytic conductor: 2.680772.68077
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 810000, ( :1/2,1/2), 1)(4,\ 810000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5415193722.541519372
L(12)L(\frac12) ≈\approx 2.5415193722.541519372
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
3 1 1
5 1 1
good7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C22C_2^2 1+23T2+p2T4 1 + 23 T^{2} + p^{2} T^{4}
17C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
19C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
23C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
37C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C22C_2^2 1+83T2+p2T4 1 + 83 T^{2} + p^{2} T^{4}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
67C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
79C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
83C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C22C_2^2 1+167T2+p2T4 1 + 167 T^{2} + p^{2} T^{4}
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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

−10.14112622647243967370640998702, −9.977823893178558580778723791723, −9.450385650233187244685527500150, −9.261099106086246421849458954556, −8.564714861902500639015066645338, −8.219346884431257290134254543856, −7.71144063614632070805021969707, −7.50897121623622279840396021121, −6.84483149746988145169386310723, −6.57066381528960798485575627732, −5.90331239581687517956669189722, −5.57020635769809955483359577206, −4.94728532785869565693476466944, −4.72051500497346810470198581000, −3.96035182333274418898081674294, −3.46402868061878143943868317573, −2.67339752721048925628385533629, −2.66349277684815050575823334823, −1.26319395930967265118682164881, −0.925200601601659279689990278024, 0.925200601601659279689990278024, 1.26319395930967265118682164881, 2.66349277684815050575823334823, 2.67339752721048925628385533629, 3.46402868061878143943868317573, 3.96035182333274418898081674294, 4.72051500497346810470198581000, 4.94728532785869565693476466944, 5.57020635769809955483359577206, 5.90331239581687517956669189722, 6.57066381528960798485575627732, 6.84483149746988145169386310723, 7.50897121623622279840396021121, 7.71144063614632070805021969707, 8.219346884431257290134254543856, 8.564714861902500639015066645338, 9.261099106086246421849458954556, 9.450385650233187244685527500150, 9.977823893178558580778723791723, 10.14112622647243967370640998702

Graph of the ZZ-function along the critical line