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Properties

Label 4-504e2-1.1-c1e2-0-15
Degree 44
Conductor 254016254016
Sign 11
Analytic cond. 16.196216.1962
Root an. cond. 2.006102.00610
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  + 5-s − 7-s − 3·9-s + 6·11-s − 6·13-s − 4·17-s + 14·19-s + 23-s + 5·25-s − 2·29-s − 10·31-s − 35-s − 12·37-s + 8·41-s + 10·43-s − 3·45-s − 8·47-s + 4·53-s + 6·55-s − 7·61-s + 3·63-s − 6·65-s + 12·67-s + 30·71-s − 4·73-s − 6·77-s − 79-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 9-s + 1.80·11-s − 1.66·13-s − 0.970·17-s + 3.21·19-s + 0.208·23-s + 25-s − 0.371·29-s − 1.79·31-s − 0.169·35-s − 1.97·37-s + 1.24·41-s + 1.52·43-s − 0.447·45-s − 1.16·47-s + 0.549·53-s + 0.809·55-s − 0.896·61-s + 0.377·63-s − 0.744·65-s + 1.46·67-s + 3.56·71-s − 0.468·73-s − 0.683·77-s − 0.112·79-s + ⋯

Functional equation

Λ(s)=(254016s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(254016s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 254016254016    =    26⋅34⋅722^{6} \cdot 3^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 16.196216.1962
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 254016, ( :1/2,1/2), 1)(4,\ 254016,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7107957901.710795790
L(12)L(\frac12) ≈\approx 1.7107957901.710795790
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+pT2 1 + p T^{2}
7C2C_2 1+T+T2 1 + T + T^{2}
good5C22C_2^2 1−T−4T2−pT3+p2T4 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4}
11C22C_2^2 1−6T+25T2−6pT3+p2T4 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4}
13C22C_2^2 1+6T+23T2+6pT3+p2T4 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4}
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
23C22C_2^2 1−T−22T2−pT3+p2T4 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4}
29C22C_2^2 1+2T−25T2+2pT3+p2T4 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+10T+69T2+10pT3+p2T4 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4}
37C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
41C22C_2^2 1−8T+23T2−8pT3+p2T4 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−10T+57T2−10pT3+p2T4 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4}
47C22C_2^2 1+8T+17T2+8pT3+p2T4 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4}
53C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
59C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
61C22C_2^2 1+7T−12T2+7pT3+p2T4 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−12T+77T2−12pT3+p2T4 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4}
71C2C_2 (1−15T+pT2)2 ( 1 - 15 T + p T^{2} )^{2}
73C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
79C22C_2^2 1+T−78T2+pT3+p2T4 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4}
83C22C_2^2 1+12T+61T2+12pT3+p2T4 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4}
89C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
97C22C_2^2 1−2T−93T2−2pT3+p2T4 1 - 2 T - 93 T^{2} - 2 p T^{3} + 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

−11.14378340108373810487657285953, −10.92216905095768275382057117270, −10.02993838421158318267995573610, −9.689927047430379837963445274130, −9.359254759688915349556621399809, −9.024934753800349899135184143228, −8.790442898467280386352089049275, −7.88063406999135904968873483830, −7.38053010498683180666534694982, −7.04079806487045421841667477210, −6.70237827658543844663769146392, −5.99138620537582391937242587918, −5.43647709877168708928633998530, −5.17418723791595700847497098426, −4.57478392809827290118619777348, −3.54373699626362443910998565605, −3.42933843704614823781603007959, −2.57894530970648102149339949371, −1.87888496616004382779534560991, −0.806745403398612268029706803062, 0.806745403398612268029706803062, 1.87888496616004382779534560991, 2.57894530970648102149339949371, 3.42933843704614823781603007959, 3.54373699626362443910998565605, 4.57478392809827290118619777348, 5.17418723791595700847497098426, 5.43647709877168708928633998530, 5.99138620537582391937242587918, 6.70237827658543844663769146392, 7.04079806487045421841667477210, 7.38053010498683180666534694982, 7.88063406999135904968873483830, 8.790442898467280386352089049275, 9.024934753800349899135184143228, 9.359254759688915349556621399809, 9.689927047430379837963445274130, 10.02993838421158318267995573610, 10.92216905095768275382057117270, 11.14378340108373810487657285953

Graph of the ZZ-function along the critical line