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Properties

Label 4-1110e2-1.1-c1e2-0-19
Degree 44
Conductor 12321001232100
Sign −1-1
Analytic cond. 78.559778.5597
Root an. cond. 2.977142.97714
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  − 2·3-s + 4-s − 4·7-s + 9-s − 2·12-s + 6·13-s + 16-s + 4·19-s + 8·21-s − 25-s + 4·27-s − 4·28-s + 8·31-s + 36-s − 12·39-s − 8·43-s − 2·48-s − 49-s + 6·52-s − 8·57-s − 8·61-s − 4·63-s + 64-s − 8·67-s − 10·73-s + 2·75-s + 4·76-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 1.51·7-s + 1/3·9-s − 0.577·12-s + 1.66·13-s + 1/4·16-s + 0.917·19-s + 1.74·21-s − 1/5·25-s + 0.769·27-s − 0.755·28-s + 1.43·31-s + 1/6·36-s − 1.92·39-s − 1.21·43-s − 0.288·48-s − 1/7·49-s + 0.832·52-s − 1.05·57-s − 1.02·61-s − 0.503·63-s + 1/8·64-s − 0.977·67-s − 1.17·73-s + 0.230·75-s + 0.458·76-s + ⋯

Functional equation

Λ(s)=(1232100s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1232100s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12321001232100    =    22⋅32⋅52⋅3722^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}
Sign: −1-1
Analytic conductor: 78.559778.5597
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 1232100, ( :1/2,1/2), −1)(4,\ 1232100,\ (\ :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)
bad2C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
3C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
5C2C_2 1+T2 1 + T^{2}
37C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good7C2C_2×\timesC2C_2 (1+T+pT2)(1+3T+pT2) ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} )
11C22C_2^2 1−7T2+p2T4 1 - 7 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−5T+pT2)(1−T+pT2) ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} )
17C22C_2^2 1−9T2+p2T4 1 - 9 T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1−5T+pT2)(1+T+pT2) ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )
23C22C_2^2 1+29T2+p2T4 1 + 29 T^{2} + p^{2} T^{4}
29C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
41C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C22C_2^2 1−62T2+p2T4 1 - 62 T^{2} + p^{2} T^{4}
53C22C_2^2 1+51T2+p2T4 1 + 51 T^{2} + p^{2} T^{4}
59C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1+2T+pT2)(1+6T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−T+pT2)(1+11T+pT2) ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−12T+pT2)(1+14T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} )
83C22C_2^2 1−7T2+p2T4 1 - 7 T^{2} + p^{2} T^{4}
89C22C_2^2 1+15T2+p2T4 1 + 15 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−6T+pT2)(1−4T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 4 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

−7.74151171551449502330125125699, −7.16152902049794544132675437202, −6.70040122798654550620733112587, −6.40818821278537747508017147952, −6.10065510704317972095508908800, −5.81156544978872936451623421426, −5.26638380188506621374707060834, −4.71491570384763710121902296680, −4.16198853833206437749890734961, −3.42665838600910385696912032166, −3.19110399319157223137105548532, −2.68985241690590522148729296526, −1.59651631208303436278229670138, −1.00402404230437196821566182156, 0, 1.00402404230437196821566182156, 1.59651631208303436278229670138, 2.68985241690590522148729296526, 3.19110399319157223137105548532, 3.42665838600910385696912032166, 4.16198853833206437749890734961, 4.71491570384763710121902296680, 5.26638380188506621374707060834, 5.81156544978872936451623421426, 6.10065510704317972095508908800, 6.40818821278537747508017147952, 6.70040122798654550620733112587, 7.16152902049794544132675437202, 7.74151171551449502330125125699

Graph of the ZZ-function along the critical line