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Properties

Label 4-1722e2-1.1-c1e2-0-5
Degree 44
Conductor 29652842965284
Sign 11
Analytic cond. 189.069189.069
Root an. cond. 3.708133.70813
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 4·5-s − 4·8-s − 9-s − 8·10-s + 5·16-s + 2·18-s + 12·20-s + 2·25-s + 4·31-s − 6·32-s − 3·36-s + 12·37-s − 16·40-s − 8·41-s + 8·43-s − 4·45-s − 49-s − 4·50-s − 12·59-s − 16·61-s − 8·62-s + 7·64-s + 4·72-s + 20·73-s − 24·74-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.78·5-s − 1.41·8-s − 1/3·9-s − 2.52·10-s + 5/4·16-s + 0.471·18-s + 2.68·20-s + 2/5·25-s + 0.718·31-s − 1.06·32-s − 1/2·36-s + 1.97·37-s − 2.52·40-s − 1.24·41-s + 1.21·43-s − 0.596·45-s − 1/7·49-s − 0.565·50-s − 1.56·59-s − 2.04·61-s − 1.01·62-s + 7/8·64-s + 0.471·72-s + 2.34·73-s − 2.78·74-s + ⋯

Functional equation

Λ(s)=(2965284s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2965284 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2965284s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2965284 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 29652842965284    =    22⋅32⋅72⋅4122^{2} \cdot 3^{2} \cdot 7^{2} \cdot 41^{2}
Sign: 11
Analytic conductor: 189.069189.069
Root analytic conductor: 3.708133.70813
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2965284, ( :1/2,1/2), 1)(4,\ 2965284,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5370849251.537084925
L(12)L(\frac12) ≈\approx 1.5370849251.537084925
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)2 ( 1 + T )^{2}
3C2C_2 1+T2 1 + T^{2}
7C2C_2 1+T2 1 + T^{2}
41C2C_2 1+8T+pT2 1 + 8 T + p T^{2}
good5C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
37C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C22C_2^2 1−78T2+p2T4 1 - 78 T^{2} + p^{2} T^{4}
53C22C_2^2 1−42T2+p2T4 1 - 42 T^{2} + p^{2} T^{4}
59C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
61C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
67C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
71C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C22C_2^2 1−142T2+p2T4 1 - 142 T^{2} + p^{2} T^{4}
83C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
89C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
97C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
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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

−9.608451289161798004818796404023, −9.264229158285421089095512436177, −8.662281026403992661469811046765, −8.644267008605309030013551729096, −7.81334148258004356158541186665, −7.80971384126690279828508104523, −7.25540427100577117569940722156, −6.71340955734571335601981998644, −6.17071680719276140727220091349, −6.12126332920449288911997929306, −5.76133762770865021089612281801, −5.23546132688697419879236468340, −4.61269392386010833313536331308, −4.17109034182542498846162728644, −3.11086484128558782310879360728, −3.06108901736996037579990096682, −2.12002710720056025871149247640, −2.06626781132466217519800232724, −1.35275979495854795980036423769, −0.60301879878664495448848779196, 0.60301879878664495448848779196, 1.35275979495854795980036423769, 2.06626781132466217519800232724, 2.12002710720056025871149247640, 3.06108901736996037579990096682, 3.11086484128558782310879360728, 4.17109034182542498846162728644, 4.61269392386010833313536331308, 5.23546132688697419879236468340, 5.76133762770865021089612281801, 6.12126332920449288911997929306, 6.17071680719276140727220091349, 6.71340955734571335601981998644, 7.25540427100577117569940722156, 7.80971384126690279828508104523, 7.81334148258004356158541186665, 8.644267008605309030013551729096, 8.662281026403992661469811046765, 9.264229158285421089095512436177, 9.608451289161798004818796404023

Graph of the ZZ-function along the critical line