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Properties

Label 4-105e2-1.1-c1e2-0-12
Degree 44
Conductor 1102511025
Sign 11
Analytic cond. 0.7029630.702963
Root an. cond. 0.9156570.915657
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4·4-s − 5-s + 9·6-s − 5·7-s − 3·8-s + 6·9-s + 3·10-s − 6·11-s − 12·12-s + 15·14-s + 3·15-s + 3·16-s − 6·17-s − 18·18-s − 12·19-s − 4·20-s + 15·21-s + 18·22-s + 3·23-s + 9·24-s − 9·27-s − 20·28-s − 9·30-s − 6·31-s − 6·32-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 0.447·5-s + 3.67·6-s − 1.88·7-s − 1.06·8-s + 2·9-s + 0.948·10-s − 1.80·11-s − 3.46·12-s + 4.00·14-s + 0.774·15-s + 3/4·16-s − 1.45·17-s − 4.24·18-s − 2.75·19-s − 0.894·20-s + 3.27·21-s + 3.83·22-s + 0.625·23-s + 1.83·24-s − 1.73·27-s − 3.77·28-s − 1.64·30-s − 1.07·31-s − 1.06·32-s + ⋯

Functional equation

Λ(s)=(11025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(11025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1102511025    =    32⋅52⋅723^{2} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 0.7029630.702963
Root analytic conductor: 0.9156570.915657
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 11025, ( :1/2,1/2), 1)(4,\ 11025,\ (\ :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)
bad3C2C_2 1+pT+pT2 1 + p T + p T^{2}
5C2C_2 1+T+T2 1 + T + T^{2}
7C2C_2 1+5T+pT2 1 + 5 T + p T^{2}
good2C22C_2^2 1+3T+5T2+3pT3+p2T4 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+6T+23T2+6pT3+p2T4 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4}
13C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
17C22C_2^2 1+6T+19T2+6pT3+p2T4 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+12T+67T2+12pT3+p2T4 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−3T+26T2−3pT3+p2T4 1 - 3 T + 26 T^{2} - 3 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−55T2+p2T4 1 - 55 T^{2} + p^{2} T^{4}
31C22C_2^2 1+6T+43T2+6pT3+p2T4 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+4T−21T2+4pT3+p2T4 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4}
41C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
43C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
59C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
61C22C_2^2 1+9T+88T2+9pT3+p2T4 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−13T+102T2−13pT3+p2T4 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−94T2+p2T4 1 - 94 T^{2} + p^{2} T^{4}
73C22C_2^2 1−6T+85T2−6pT3+p2T4 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−16T+177T2−16pT3+p2T4 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4}
83C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
89C22C_2^2 1+3T−80T2+3pT3+p2T4 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−86T2+p2T4 1 - 86 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

−12.86030469335877291228631811314, −12.81278060750407138636131065364, −12.70460821430385747282035445610, −11.68384543798572901545518948867, −10.88671961733396961010145934687, −10.79937734987930125606157001943, −10.20478907525436304041427489254, −10.07409954279382640044032862993, −8.983541384324832939876145785173, −8.974408970577501580761814618917, −8.102770939601159033698036488876, −7.43357468040600120656989299546, −6.59014411661458313488026358895, −6.57616908830316756187719347272, −5.65540329908988067541756759561, −4.82403773402819725151125010044, −3.81214811365540460137061738021, −2.37166676432777566319260691004, 0, 0, 2.37166676432777566319260691004, 3.81214811365540460137061738021, 4.82403773402819725151125010044, 5.65540329908988067541756759561, 6.57616908830316756187719347272, 6.59014411661458313488026358895, 7.43357468040600120656989299546, 8.102770939601159033698036488876, 8.974408970577501580761814618917, 8.983541384324832939876145785173, 10.07409954279382640044032862993, 10.20478907525436304041427489254, 10.79937734987930125606157001943, 10.88671961733396961010145934687, 11.68384543798572901545518948867, 12.70460821430385747282035445610, 12.81278060750407138636131065364, 12.86030469335877291228631811314

Graph of the ZZ-function along the critical line