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Properties

Label 4-740772-1.1-c1e2-0-8
Degree 44
Conductor 740772740772
Sign −1-1
Analytic cond. 47.232247.2322
Root an. cond. 2.621552.62155
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s + 9-s + 12-s + 2·13-s + 16-s + 19-s + 6·25-s − 27-s + 2·31-s − 36-s + 6·37-s − 2·39-s − 8·43-s − 48-s − 10·49-s − 2·52-s − 57-s − 12·61-s − 64-s − 16·67-s − 16·73-s − 6·75-s − 76-s + 2·79-s + 81-s − 2·93-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.229·19-s + 6/5·25-s − 0.192·27-s + 0.359·31-s − 1/6·36-s + 0.986·37-s − 0.320·39-s − 1.21·43-s − 0.144·48-s − 1.42·49-s − 0.277·52-s − 0.132·57-s − 1.53·61-s − 1/8·64-s − 1.95·67-s − 1.87·73-s − 0.692·75-s − 0.114·76-s + 0.225·79-s + 1/9·81-s − 0.207·93-s + ⋯

Functional equation

Λ(s)=(740772s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(740772s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 740772740772    =    22⋅33⋅1932^{2} \cdot 3^{3} \cdot 19^{3}
Sign: −1-1
Analytic conductor: 47.232247.2322
Root analytic conductor: 2.621552.62155
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 740772, ( :1/2,1/2), −1)(4,\ 740772,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2C2C_2 1+T2 1 + T^{2}
3C1C_1 1+T 1 + T
19C1C_1 1−T 1 - T
good5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−2T+pT2)(1+pT2) ( 1 - 2 T + p T^{2} )( 1 + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
29C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
31C2C_2×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
37C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−4T+pT2)(1+12T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C22C_2^2 1+42T2+p2T4 1 + 42 T^{2} + p^{2} T^{4}
53C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
59C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1+4T+pT2)(1+8T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C2C_2×\timesC2C_2 (1+4T+pT2)(1+12T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C22C_2^2 1−62T2+p2T4 1 - 62 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1+6T+pT2)(1+10T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−6T+pT2)(1+4T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
89C22C_2^2 1−110T2+p2T4 1 - 110 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1+6T+pT2)(1+14T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 14 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

−8.183543219577224388436904152477, −7.63630335589458345937232249879, −7.14083961353630804979395184532, −6.63476570475728421128822946853, −6.28451854901339919449557531977, −5.77763744585875121435967355110, −5.34110224567434626668185813894, −4.74329972392802860888192543540, −4.46245327973518306899198622211, −3.92046810864675229498181414898, −3.07364753727346617579683991524, −2.90060183359140465290290245010, −1.68407289406377645559105544028, −1.15856078640089366274780288036, 0, 1.15856078640089366274780288036, 1.68407289406377645559105544028, 2.90060183359140465290290245010, 3.07364753727346617579683991524, 3.92046810864675229498181414898, 4.46245327973518306899198622211, 4.74329972392802860888192543540, 5.34110224567434626668185813894, 5.77763744585875121435967355110, 6.28451854901339919449557531977, 6.63476570475728421128822946853, 7.14083961353630804979395184532, 7.63630335589458345937232249879, 8.183543219577224388436904152477

Graph of the ZZ-function along the critical line