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Properties

Label 4-229e2-1.1-c1e2-0-1
Degree 44
Conductor 5244152441
Sign 11
Analytic cond. 3.343683.34368
Root an. cond. 1.352241.35224
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  + 2·3-s − 4-s + 6·5-s − 3·9-s + 6·11-s − 2·12-s + 12·15-s − 3·16-s − 6·17-s − 2·19-s − 6·20-s + 17·25-s − 14·27-s + 12·33-s + 3·36-s + 4·37-s − 2·43-s − 6·44-s − 18·45-s − 6·48-s + 14·49-s − 12·51-s + 12·53-s + 36·55-s − 4·57-s − 12·60-s + 10·61-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 2.68·5-s − 9-s + 1.80·11-s − 0.577·12-s + 3.09·15-s − 3/4·16-s − 1.45·17-s − 0.458·19-s − 1.34·20-s + 17/5·25-s − 2.69·27-s + 2.08·33-s + 1/2·36-s + 0.657·37-s − 0.304·43-s − 0.904·44-s − 2.68·45-s − 0.866·48-s + 2·49-s − 1.68·51-s + 1.64·53-s + 4.85·55-s − 0.529·57-s − 1.54·60-s + 1.28·61-s + ⋯

Functional equation

Λ(s)=(52441s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 52441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(52441s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 52441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5244152441    =    2292229^{2}
Sign: 11
Analytic conductor: 3.343683.34368
Root analytic conductor: 1.352241.35224
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 52441, ( :1/2,1/2), 1)(4,\ 52441,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5495810912.549581091
L(12)L(\frac12) ≈\approx 2.5495810912.549581091
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)
bad229C2C_2 1−14T+pT2 1 - 14 T + p T^{2}
good2C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
3C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
5C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
7C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
13C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
19C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
23C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
29C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
31C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
43C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
47C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
61C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
67C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
71C2C_2 (1+15T+pT2)2 ( 1 + 15 T + p T^{2} )^{2}
73C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
79C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
83C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1+7T+pT2)2 ( 1 + 7 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

−13.14580004509351118819641553580, −11.75485079284358130848965550474, −11.61430493196621781706902305185, −10.99769158155824883235202256723, −10.20704899738732125371168016271, −9.933228883600645345117312716392, −9.288995754727251390007447684528, −8.998106452521941953152706182178, −8.768961470526460026351817099329, −8.587851934459013834761684572790, −7.41739122861685143280509277723, −6.71778066430661336888582716183, −6.29205951881109080234144756635, −5.78640405170751959347105161981, −5.42927634164617286025699613855, −4.38492488245376512643072179509, −3.88261708380527167211809567495, −2.66907075111205383917085810778, −2.37938663941908880657930341002, −1.65730355152551961546014863016, 1.65730355152551961546014863016, 2.37938663941908880657930341002, 2.66907075111205383917085810778, 3.88261708380527167211809567495, 4.38492488245376512643072179509, 5.42927634164617286025699613855, 5.78640405170751959347105161981, 6.29205951881109080234144756635, 6.71778066430661336888582716183, 7.41739122861685143280509277723, 8.587851934459013834761684572790, 8.768961470526460026351817099329, 8.998106452521941953152706182178, 9.288995754727251390007447684528, 9.933228883600645345117312716392, 10.20704899738732125371168016271, 10.99769158155824883235202256723, 11.61430493196621781706902305185, 11.75485079284358130848965550474, 13.14580004509351118819641553580

Graph of the ZZ-function along the critical line