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Properties

Label 4-243675-1.1-c1e2-0-5
Degree 44
Conductor 243675243675
Sign −1-1
Analytic cond. 15.536915.5369
Root an. cond. 1.985361.98536
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s + 4·7-s + 9-s + 12-s + 2·13-s − 3·16-s − 6·19-s − 4·21-s − 25-s − 27-s − 4·28-s − 2·31-s − 36-s − 4·37-s − 2·39-s − 10·43-s + 3·48-s + 2·49-s − 2·52-s + 6·57-s − 4·61-s + 4·63-s + 7·64-s − 6·67-s + 75-s + 6·76-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s + 1.51·7-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 3/4·16-s − 1.37·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s − 0.755·28-s − 0.359·31-s − 1/6·36-s − 0.657·37-s − 0.320·39-s − 1.52·43-s + 0.433·48-s + 2/7·49-s − 0.277·52-s + 0.794·57-s − 0.512·61-s + 0.503·63-s + 7/8·64-s − 0.733·67-s + 0.115·75-s + 0.688·76-s + ⋯

Functional equation

Λ(s)=(243675s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(243675s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 243675243675    =    33⋅52⋅1923^{3} \cdot 5^{2} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 15.536915.5369
Root analytic conductor: 1.985361.98536
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 243675, ( :1/2,1/2), −1)(4,\ 243675,\ (\ :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)
bad3C1C_1 1+T 1 + T
5C2C_2 1+T2 1 + T^{2}
19C2C_2 1+6T+pT2 1 + 6 T + p T^{2}
good2C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
7C2C_2×\timesC2C_2 (1−4T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
11C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
23C22C_2^2 1−40T2+p2T4 1 - 40 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2×\timesC2C_2 (1+pT2)(1+2T+pT2) ( 1 + p T^{2} )( 1 + 2 T + p T^{2} )
37C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C22C_2^2 1+32T2+p2T4 1 + 32 T^{2} + p^{2} T^{4}
53C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
59C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−4T+pT2)(1+10T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
71C22C_2^2 1−90T2+p2T4 1 - 90 T^{2} + p^{2} T^{4}
73C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
79C2C_2×\timesC2C_2 (1+8T+pT2)(1+12T+pT2) ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} )
83C22C_2^2 1−44T2+p2T4 1 - 44 T^{2} + p^{2} T^{4}
89C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1+8T+pT2)(1+10T+pT2) ( 1 + 8 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

−8.593917946955164955871663391398, −8.409050662816037013658497211522, −7.85668232398436390657866215570, −7.36340252768348613997453202888, −6.59003416492810382038434874830, −6.53512873780771400495109952978, −5.57075046284373331718582908483, −5.36004521642751391049304360231, −4.62388312892885559329095659142, −4.39223064490415808368029175552, −3.85023417367922247417552763507, −2.92400971955770211327520464920, −1.92577471099237739968377247932, −1.48413346507443369161719444742, 0, 1.48413346507443369161719444742, 1.92577471099237739968377247932, 2.92400971955770211327520464920, 3.85023417367922247417552763507, 4.39223064490415808368029175552, 4.62388312892885559329095659142, 5.36004521642751391049304360231, 5.57075046284373331718582908483, 6.53512873780771400495109952978, 6.59003416492810382038434874830, 7.36340252768348613997453202888, 7.85668232398436390657866215570, 8.409050662816037013658497211522, 8.593917946955164955871663391398

Graph of the ZZ-function along the critical line