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Properties

Label 4-43904-1.1-c1e2-0-7
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 yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s + 2·9-s − 14-s + 16-s + 2·18-s + 10·25-s − 28-s + 32-s + 2·36-s + 49-s + 10·50-s − 56-s − 2·63-s + 64-s + 2·72-s − 16·79-s − 5·81-s + 98-s + 10·100-s − 112-s − 12·113-s − 22·121-s − 2·126-s + 127-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 2/3·9-s − 0.267·14-s + 1/4·16-s + 0.471·18-s + 2·25-s − 0.188·28-s + 0.176·32-s + 1/3·36-s + 1/7·49-s + 1.41·50-s − 0.133·56-s − 0.251·63-s + 1/8·64-s + 0.235·72-s − 1.80·79-s − 5/9·81-s + 0.101·98-s + 100-s − 0.0944·112-s − 1.12·113-s − 2·121-s − 0.178·126-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: yes
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 2.1056856362.105685636
L(12)L(\frac12) ≈\approx 2.1056856362.105685636
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
good3C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
5C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{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)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 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 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
61C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
67C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 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

−10.35429207920592299666236056630, −9.737571352224082577057668604337, −9.230234198592328844814622223277, −8.647620969069333470108951478696, −8.102096221801790937263883784465, −7.36010673208194959503095702760, −6.94438036583863366124517437556, −6.51987156391339999128357750064, −5.82258912862470940314930166189, −5.18598252263235902284418013548, −4.59160399106119589499226900700, −4.00728883804981885022921502277, −3.20128729290109899034222112750, −2.55522247581925182860702086473, −1.34090643042552658976895105787, 1.34090643042552658976895105787, 2.55522247581925182860702086473, 3.20128729290109899034222112750, 4.00728883804981885022921502277, 4.59160399106119589499226900700, 5.18598252263235902284418013548, 5.82258912862470940314930166189, 6.51987156391339999128357750064, 6.94438036583863366124517437556, 7.36010673208194959503095702760, 8.102096221801790937263883784465, 8.647620969069333470108951478696, 9.230234198592328844814622223277, 9.737571352224082577057668604337, 10.35429207920592299666236056630

Graph of the ZZ-function along the critical line