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Properties

Label 4-665e2-1.1-c1e2-0-1
Degree 44
Conductor 442225442225
Sign 11
Analytic cond. 28.196628.1966
Root an. cond. 2.304352.30435
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-s − 2·3-s + 2·4-s + 5-s − 2·6-s − 4·7-s + 5·8-s − 3·9-s + 10-s − 3·11-s − 4·12-s − 5·13-s − 4·14-s − 2·15-s + 5·16-s − 6·17-s − 3·18-s + 7·19-s + 2·20-s + 8·21-s − 3·22-s − 14·23-s − 10·24-s − 5·26-s + 14·27-s − 8·28-s − 7·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 4-s + 0.447·5-s − 0.816·6-s − 1.51·7-s + 1.76·8-s − 9-s + 0.316·10-s − 0.904·11-s − 1.15·12-s − 1.38·13-s − 1.06·14-s − 0.516·15-s + 5/4·16-s − 1.45·17-s − 0.707·18-s + 1.60·19-s + 0.447·20-s + 1.74·21-s − 0.639·22-s − 2.91·23-s − 2.04·24-s − 0.980·26-s + 2.69·27-s − 1.51·28-s − 1.29·29-s + ⋯

Functional equation

Λ(s)=(442225s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(442225s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 442225442225    =    52⋅72⋅1925^{2} \cdot 7^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 28.196628.1966
Root analytic conductor: 2.304352.30435
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 442225, ( :1/2,1/2), 1)(4,\ 442225,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.82272460140.8227246014
L(12)L(\frac12) ≈\approx 0.82272460140.8227246014
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)
bad5C2C_2 1−T+T2 1 - T + T^{2}
7C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
19C2C_2 1−7T+pT2 1 - 7 T + p T^{2}
good2C22C_2^2 1−T−T2−pT3+p2T4 1 - T - T^{2} - p T^{3} + p^{2} T^{4}
3C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
11C22C_2^2 1+3T−2T2+3pT3+p2T4 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4}
13C2C_2 (1−2T+pT2)(1+7T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} )
17C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
23C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
29C22C_2^2 1+7T+20T2+7pT3+p2T4 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4}
31C22C_2^2 1−5T−6T2−5pT3+p2T4 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4}
37C22C_2^2 1−3T−28T2−3pT3+p2T4 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4}
41C22C_2^2 1−T−40T2−pT3+p2T4 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4}
43C22C_2^2 1−11T+78T2−11pT3+p2T4 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4}
47C2C_2 (1+11T+pT2)2 ( 1 + 11 T + p T^{2} )^{2}
53C22C_2^2 1−10T+47T2−10pT3+p2T4 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4}
59C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
61C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
67C22C_2^2 1+12T+77T2+12pT3+p2T4 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−T−70T2−pT3+p2T4 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4}
73C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
79C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
83C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
89C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
97C2C_2 (1−14T+pT2)(1+19T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 19 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

−11.04810506722079392610604168601, −10.28319571736789455632240517658, −9.872036998041582077472852435303, −9.839595035518194324831741362933, −9.273232199948768071947077071139, −8.268828255590503060168776330654, −8.145648580987681945901113129016, −7.44571934435379874629893617307, −7.07312929641254898814967058733, −6.50941044628014587976451666950, −6.23406030329819193729561259224, −5.79013511825916754156805582084, −5.26597203143933976397524961780, −5.11333476582095494855558285947, −4.29756212507897062564521025234, −3.77977546925852383595195678130, −2.87991705554582719859113337328, −2.51173710233023751336438617104, −2.03822747743073103949035430428, −0.42499045854351675994417798783, 0.42499045854351675994417798783, 2.03822747743073103949035430428, 2.51173710233023751336438617104, 2.87991705554582719859113337328, 3.77977546925852383595195678130, 4.29756212507897062564521025234, 5.11333476582095494855558285947, 5.26597203143933976397524961780, 5.79013511825916754156805582084, 6.23406030329819193729561259224, 6.50941044628014587976451666950, 7.07312929641254898814967058733, 7.44571934435379874629893617307, 8.145648580987681945901113129016, 8.268828255590503060168776330654, 9.273232199948768071947077071139, 9.839595035518194324831741362933, 9.872036998041582077472852435303, 10.28319571736789455632240517658, 11.04810506722079392610604168601

Graph of the ZZ-function along the critical line