Properties

Label 4-395136-1.1-c1e2-0-5
Degree 44
Conductor 395136395136
Sign 11
Analytic cond. 25.194225.1942
Root an. cond. 2.240392.24039
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  − 7-s − 9-s + 2·11-s + 6·23-s − 2·25-s − 4·29-s − 4·37-s + 4·43-s + 49-s + 8·53-s + 63-s + 16·67-s − 6·71-s − 2·77-s + 16·79-s + 81-s − 2·99-s − 6·107-s − 20·109-s − 4·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.377·7-s − 1/3·9-s + 0.603·11-s + 1.25·23-s − 2/5·25-s − 0.742·29-s − 0.657·37-s + 0.609·43-s + 1/7·49-s + 1.09·53-s + 0.125·63-s + 1.95·67-s − 0.712·71-s − 0.227·77-s + 1.80·79-s + 1/9·81-s − 0.201·99-s − 0.580·107-s − 1.91·109-s − 0.376·113-s − 0.909·121-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 + ⋯

Functional equation

Λ(s)=(395136s/2ΓC(s)2L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(395136s/2ΓC(s+1/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 395136395136    =    2732732^{7} \cdot 3^{2} \cdot 7^{3}
Sign: 11
Analytic conductor: 25.194225.1942
Root analytic conductor: 2.240392.24039
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 395136, ( :1/2,1/2), 1)(4,\ 395136,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) \approx 1.6340754981.634075498
L(12)L(\frac12) \approx 1.6340754981.634075498
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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+T2 1 + T^{2}
7C1C_1 1+T 1 + T
good5C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2×\timesC2C_2 (14T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
13C2C_2 (16T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
19C2C_2 (14T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2×\timesC2C_2 (18T+pT2)(1+2T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} )
29C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
31C2C_2 (110T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
37C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
41C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (14T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
47C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
53C2C_2×\timesC2C_2 (16T+pT2)(12T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
59C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
61C22C_2^2 142T2+p2T4 1 - 42 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (112T+pT2)(14T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )
71C2C_2×\timesC2C_2 (16T+pT2)(1+12T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C22C_2^2 1+78T2+p2T4 1 + 78 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (116T+pT2)(1+pT2) ( 1 - 16 T + p T^{2} )( 1 + p T^{2} )
83C22C_2^2 186T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
89C22C_2^2 1+42T2+p2T4 1 + 42 T^{2} + p^{2} T^{4}
97C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
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   L(s)=p j=14(1αj,pps)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

−8.699282088321589201467643517093, −8.236972050375983718950285796462, −7.77018949902582179915957339205, −7.14070251885670841829509075166, −6.87700863147116782395497369892, −6.39344426492772010571509299168, −5.80637317277794281955672583724, −5.36007472380367461897506998623, −4.92880193215004895162898338699, −4.11584250552478313262890669286, −3.76479384395257081325203691637, −3.13250433380878648175129575563, −2.49956093243554054195671052448, −1.73125840677039039003718737465, −0.72046474937322208138838996608, 0.72046474937322208138838996608, 1.73125840677039039003718737465, 2.49956093243554054195671052448, 3.13250433380878648175129575563, 3.76479384395257081325203691637, 4.11584250552478313262890669286, 4.92880193215004895162898338699, 5.36007472380367461897506998623, 5.80637317277794281955672583724, 6.39344426492772010571509299168, 6.87700863147116782395497369892, 7.14070251885670841829509075166, 7.77018949902582179915957339205, 8.236972050375983718950285796462, 8.699282088321589201467643517093

Graph of the ZZ-function along the critical line