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Properties

Label 4-1408e2-1.1-c1e2-0-26
Degree 44
Conductor 19824641982464
Sign 11
Analytic cond. 126.403126.403
Root an. cond. 3.353043.35304
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  + 2·3-s − 2·5-s − 9-s − 2·11-s − 4·15-s − 12·17-s + 4·19-s − 14·23-s + 25-s − 6·27-s − 16·29-s − 2·31-s − 4·33-s + 6·37-s + 4·43-s + 2·45-s − 8·47-s − 12·49-s − 24·51-s + 4·55-s + 8·57-s + 2·59-s + 6·67-s − 28·69-s − 2·71-s − 16·73-s + 2·75-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 1/3·9-s − 0.603·11-s − 1.03·15-s − 2.91·17-s + 0.917·19-s − 2.91·23-s + 1/5·25-s − 1.15·27-s − 2.97·29-s − 0.359·31-s − 0.696·33-s + 0.986·37-s + 0.609·43-s + 0.298·45-s − 1.16·47-s − 1.71·49-s − 3.36·51-s + 0.539·55-s + 1.05·57-s + 0.260·59-s + 0.733·67-s − 3.37·69-s − 0.237·71-s − 1.87·73-s + 0.230·75-s + ⋯

Functional equation

Λ(s)=(1982464s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1982464s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 19824641982464    =    214⋅1122^{14} \cdot 11^{2}
Sign: 11
Analytic conductor: 126.403126.403
Root analytic conductor: 3.353043.35304
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 1982464, ( :1/2,1/2), 1)(4,\ 1982464,\ (\ :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)
bad2 1 1
11C1C_1 (1+T)2 ( 1 + T )^{2}
good3D4D_{4} 1−2T+5T2−2pT3+p2T4 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4}
5D4D_{4} 1+2T+3T2+2pT3+p2T4 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4}
7C22C_2^2 1+12T2+p2T4 1 + 12 T^{2} + p^{2} T^{4}
13C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
17D4D_{4} 1+12T+4pT2+12pT3+p2T4 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4}
19D4D_{4} 1−4T+40T2−4pT3+p2T4 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+14T+93T2+14pT3+p2T4 1 + 14 T + 93 T^{2} + 14 p T^{3} + p^{2} T^{4}
29C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
31D4D_{4} 1+2T+13T2+2pT3+p2T4 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−6T+75T2−6pT3+p2T4 1 - 6 T + 75 T^{2} - 6 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+32T2+p2T4 1 + 32 T^{2} + p^{2} T^{4}
43D4D_{4} 1−4T+82T2−4pT3+p2T4 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+8T+38T2+8pT3+p2T4 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+98T2+p2T4 1 + 98 T^{2} + p^{2} T^{4}
59D4D_{4} 1−2T−43T2−2pT3+p2T4 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+90T2+p2T4 1 + 90 T^{2} + p^{2} T^{4}
67D4D_{4} 1−6T−19T2−6pT3+p2T4 1 - 6 T - 19 T^{2} - 6 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+2T+93T2+2pT3+p2T4 1 + 2 T + 93 T^{2} + 2 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+16T+160T2+16pT3+p2T4 1 + 16 T + 160 T^{2} + 16 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+4T+154T2+4pT3+p2T4 1 + 4 T + 154 T^{2} + 4 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−28T+354T2−28pT3+p2T4 1 - 28 T + 354 T^{2} - 28 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+2T+171T2+2pT3+p2T4 1 + 2 T + 171 T^{2} + 2 p T^{3} + p^{2} T^{4}
97C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{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

−9.307792892000514667433720601060, −8.941278534885239021127729197559, −8.323259246311900458736051288468, −8.317775442759805899000068256204, −7.74507549059206434494460567708, −7.62390622147860683081293782265, −7.06866181070146480199960466027, −6.55334978843337743957316088391, −5.96557144008317646980639787980, −5.77572163956453322882330457866, −5.09578898975605732472636407584, −4.47180994629019935382320498234, −4.14875096834245229596544178879, −3.67772654398545225543015562536, −3.30253751061432357148054251435, −2.62960626220228083891303419450, −1.98358057460718470239393544702, −1.96833375484122923003211673218, 0, 0, 1.96833375484122923003211673218, 1.98358057460718470239393544702, 2.62960626220228083891303419450, 3.30253751061432357148054251435, 3.67772654398545225543015562536, 4.14875096834245229596544178879, 4.47180994629019935382320498234, 5.09578898975605732472636407584, 5.77572163956453322882330457866, 5.96557144008317646980639787980, 6.55334978843337743957316088391, 7.06866181070146480199960466027, 7.62390622147860683081293782265, 7.74507549059206434494460567708, 8.317775442759805899000068256204, 8.323259246311900458736051288468, 8.941278534885239021127729197559, 9.307792892000514667433720601060

Graph of the ZZ-function along the critical line