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Properties

Label 4-243675-1.1-c1e2-0-4
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 − 2·4-s + 9-s − 2·12-s − 2·13-s + 19-s − 25-s + 27-s + 3·31-s − 2·36-s − 4·37-s − 2·39-s − 10·43-s − 10·49-s + 4·52-s + 57-s − 2·61-s + 8·64-s + 8·67-s − 8·73-s − 75-s − 2·76-s + 81-s + 3·93-s − 14·97-s + 2·100-s − 2·108-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 1/3·9-s − 0.577·12-s − 0.554·13-s + 0.229·19-s − 1/5·25-s + 0.192·27-s + 0.538·31-s − 1/3·36-s − 0.657·37-s − 0.320·39-s − 1.52·43-s − 1.42·49-s + 0.554·52-s + 0.132·57-s − 0.256·61-s + 64-s + 0.977·67-s − 0.936·73-s − 0.115·75-s − 0.229·76-s + 1/9·81-s + 0.311·93-s − 1.42·97-s + 1/5·100-s − 0.192·108-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−T+pT2 1 - T + p T^{2}
good2C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C22C_2^2 1−9T2+p2T4 1 - 9 T^{2} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)(1+6T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1−19T2+p2T4 1 - 19 T^{2} + p^{2} T^{4}
23C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
29C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
31C2C_2×\timesC2C_2 (1−3T+pT2)(1+pT2) ( 1 - 3 T + p T^{2} )( 1 + p T^{2} )
37C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+4T+pT2)(1+6T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
47C22C_2^2 1+71T2+p2T4 1 + 71 T^{2} + p^{2} T^{4}
53C22C_2^2 1+19T2+p2T4 1 + 19 T^{2} + p^{2} T^{4}
59C22C_2^2 1+61T2+p2T4 1 + 61 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−11T+pT2)(1+13T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
71C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−8T+pT2)(1+16T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
83C22C_2^2 1−93T2+p2T4 1 - 93 T^{2} + p^{2} T^{4}
89C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−4T+pT2)(1+18T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 18 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

−8.633630525341493583638457503551, −8.395166941405474152027575516190, −7.914872422762183229290916967524, −7.41631768031240655787744141966, −6.80636502565617265499840869515, −6.47968958834526934246222304501, −5.69811719282712620007338569642, −5.05937629246028149837748720062, −4.81066540146337213509272971625, −4.17297521159942847561581740747, −3.61387392376109778270454534270, −3.01848660130097057822476697468, −2.27584932677099814084676102753, −1.39228421435216690840591183904, 0, 1.39228421435216690840591183904, 2.27584932677099814084676102753, 3.01848660130097057822476697468, 3.61387392376109778270454534270, 4.17297521159942847561581740747, 4.81066540146337213509272971625, 5.05937629246028149837748720062, 5.69811719282712620007338569642, 6.47968958834526934246222304501, 6.80636502565617265499840869515, 7.41631768031240655787744141966, 7.914872422762183229290916967524, 8.395166941405474152027575516190, 8.633630525341493583638457503551

Graph of the ZZ-function along the critical line