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Properties

Label 4-43904-1.1-c1e2-0-11
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 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 7-s + 6·9-s − 4·19-s + 4·21-s + 6·25-s − 4·27-s + 4·29-s + 8·31-s − 12·37-s − 8·47-s + 49-s − 20·53-s − 16·57-s + 12·59-s + 6·63-s + 24·75-s − 37·81-s + 12·83-s + 16·87-s + 32·93-s − 24·103-s + 20·109-s − 48·111-s + 12·113-s − 22·121-s + 127-s + ⋯
L(s)  = 1  + 2.30·3-s + 0.377·7-s + 2·9-s − 0.917·19-s + 0.872·21-s + 6/5·25-s − 0.769·27-s + 0.742·29-s + 1.43·31-s − 1.97·37-s − 1.16·47-s + 1/7·49-s − 2.74·53-s − 2.11·57-s + 1.56·59-s + 0.755·63-s + 2.77·75-s − 4.11·81-s + 1.31·83-s + 1.71·87-s + 3.31·93-s − 2.36·103-s + 1.91·109-s − 4.55·111-s + 1.12·113-s − 2·121-s + 0.0887·127-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: 00
Selberg data: (4, 43904, ( :1/2,1/2), 1)(4,\ 43904,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7175350602.717535060
L(12)L(\frac12) ≈\approx 2.7175350602.717535060
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
7C1C_1 1−T 1 - T
good3C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
37C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
41C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
43C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
53C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
59C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
61C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
67C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
89C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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

−9.835763078329112662938551691511, −9.676028605922657643630885041741, −8.928329697232844766585907221767, −8.589598276995096865608741769842, −8.188057214509100026281306554311, −8.037894346653340821155380595819, −7.14438434722230698367292292955, −6.69299348503088401254902880103, −5.96146004102990052123544748972, −4.99083605129769608898331014735, −4.50185552371175935345657331179, −3.56791835941750047835108384805, −3.14137792992390816749783057891, −2.48337724162238959661867349800, −1.73163501201493931998851829334, 1.73163501201493931998851829334, 2.48337724162238959661867349800, 3.14137792992390816749783057891, 3.56791835941750047835108384805, 4.50185552371175935345657331179, 4.99083605129769608898331014735, 5.96146004102990052123544748972, 6.69299348503088401254902880103, 7.14438434722230698367292292955, 8.037894346653340821155380595819, 8.188057214509100026281306554311, 8.589598276995096865608741769842, 8.928329697232844766585907221767, 9.676028605922657643630885041741, 9.835763078329112662938551691511

Graph of the ZZ-function along the critical line