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Properties

Label 4-43904-1.1-c1e2-0-16
Degree 44
Conductor 4390443904
Sign 11
Analytic cond. 2.799352.79935
Root an. cond. 1.293491.29349
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4·3-s + 4-s − 2·5-s + 4·6-s + 7-s − 8-s + 6·9-s + 2·10-s − 4·11-s − 4·12-s − 10·13-s − 14-s + 8·15-s + 16-s + 2·17-s − 6·18-s − 4·19-s − 2·20-s − 4·21-s + 4·22-s − 4·23-s + 4·24-s − 6·25-s + 10·26-s + 4·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s + 1/2·4-s − 0.894·5-s + 1.63·6-s + 0.377·7-s − 0.353·8-s + 2·9-s + 0.632·10-s − 1.20·11-s − 1.15·12-s − 2.77·13-s − 0.267·14-s + 2.06·15-s + 1/4·16-s + 0.485·17-s − 1.41·18-s − 0.917·19-s − 0.447·20-s − 0.872·21-s + 0.852·22-s − 0.834·23-s + 0.816·24-s − 6/5·25-s + 1.96·26-s + 0.769·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(43904s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(43904s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4390443904    =    27⋅732^{7} \cdot 7^{3}
Sign: 11
Analytic conductor: 2.799352.79935
Root analytic conductor: 1.293491.29349
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 43904, ( :1/2,1/2), 1)(4,\ 43904,\ (\ :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)
bad2C1C_1 1+T 1 + T
7C1C_1 1−T 1 - T
good3C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
5C2C_2×\timesC2C_2 (1+pT2)(1+2T+pT2) ( 1 + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1+4T+pT2)(1+6T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2×\timesC2C_2 (1−6T+pT2)(1+4T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )
19C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−6T+pT2)(1−4T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−8T+pT2)(1+12T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
61C2C_2×\timesC2C_2 (1−8T+pT2)(1−6T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 6 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} )
71C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
73C2C_2×\timesC2C_2 (1−2T+pT2)(1+pT2) ( 1 - 2 T + p T^{2} )( 1 + p T^{2} )
79C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
83C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
89C2C_2×\timesC2C_2 (1+6T+pT2)(1+16T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} )
97C2C_2×\timesC2C_2 (1+10T+pT2)(1+12T+pT2) ( 1 + 10 T + p T^{2} )( 1 + 12 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

−15.4514933430, −14.9728953745, −14.6077730657, −14.2451399800, −13.1881895766, −12.6417026271, −12.3052609901, −12.1063406379, −11.4418273496, −11.2313614144, −10.9409991928, −10.2459627166, −9.76554711946, −9.63516922865, −8.37116758932, −8.01003490126, −7.57571100089, −7.08238367024, −6.45217559209, −5.59673699857, −5.57928681743, −4.69839552642, −4.43201465917, −3.00626769255, −2.08688930207, 0, 0, 2.08688930207, 3.00626769255, 4.43201465917, 4.69839552642, 5.57928681743, 5.59673699857, 6.45217559209, 7.08238367024, 7.57571100089, 8.01003490126, 8.37116758932, 9.63516922865, 9.76554711946, 10.2459627166, 10.9409991928, 11.2313614144, 11.4418273496, 12.1063406379, 12.3052609901, 12.6417026271, 13.1881895766, 14.2451399800, 14.6077730657, 14.9728953745, 15.4514933430

Graph of the ZZ-function along the critical line