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Properties

Label 4-1216e2-1.1-c1e2-0-15
Degree 44
Conductor 14786561478656
Sign 11
Analytic cond. 94.280394.2803
Root an. cond. 3.116053.11605
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  + 2·5-s − 2·9-s + 8·13-s + 10·17-s − 7·25-s − 4·29-s + 20·37-s + 12·41-s − 4·45-s − 5·49-s + 16·53-s + 10·61-s + 16·65-s − 30·73-s − 5·81-s + 20·85-s + 32·97-s + 36·101-s − 24·109-s + 4·113-s − 16·117-s − 13·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 2/3·9-s + 2.21·13-s + 2.42·17-s − 7/5·25-s − 0.742·29-s + 3.28·37-s + 1.87·41-s − 0.596·45-s − 5/7·49-s + 2.19·53-s + 1.28·61-s + 1.98·65-s − 3.51·73-s − 5/9·81-s + 2.16·85-s + 3.24·97-s + 3.58·101-s − 2.29·109-s + 0.376·113-s − 1.47·117-s − 1.18·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

Λ(s)=(1478656s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1478656s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14786561478656    =    212⋅1922^{12} \cdot 19^{2}
Sign: 11
Analytic conductor: 94.280394.2803
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1478656, ( :1/2,1/2), 1)(4,\ 1478656,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3972223793.397222379
L(12)L(\frac12) ≈\approx 3.3972223793.397222379
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
19C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
5C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
7C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
13C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
17C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
47C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
53C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
59C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
61C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
67C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
71C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
73C2C_2 (1+15T+pT2)2 ( 1 + 15 T + p T^{2} )^{2}
79C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1−16T+pT2)2 ( 1 - 16 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

−8.024247643273632491999690471751, −7.43537131861627444263317650332, −7.30495770738943457952049118581, −6.20879084669067007942487517486, −5.95681555667055397919593180547, −5.89904763388088518363904025264, −5.67735294850741520947496000769, −4.88105113826037271671762872708, −4.19084046620099313866498183608, −3.73244909832136496864835491552, −3.41145508666670040670114085640, −2.67622771685970545269516699451, −2.20146006630293663601866411040, −1.26674131115266326801058270565, −0.944005207983174855702403672277, 0.944005207983174855702403672277, 1.26674131115266326801058270565, 2.20146006630293663601866411040, 2.67622771685970545269516699451, 3.41145508666670040670114085640, 3.73244909832136496864835491552, 4.19084046620099313866498183608, 4.88105113826037271671762872708, 5.67735294850741520947496000769, 5.89904763388088518363904025264, 5.95681555667055397919593180547, 6.20879084669067007942487517486, 7.30495770738943457952049118581, 7.43537131861627444263317650332, 8.024247643273632491999690471751

Graph of the ZZ-function along the critical line