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Properties

Label 4-855e2-1.1-c2e2-0-0
Degree 44
Conductor 731025731025
Sign 11
Analytic cond. 542.753542.753
Root an. cond. 4.826704.82670
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s + 9·5-s + 6·11-s + 48·16-s − 38·19-s − 72·20-s + 56·25-s − 48·44-s − 73·49-s + 54·55-s − 206·61-s − 256·64-s + 304·76-s + 432·80-s − 342·95-s − 448·100-s − 204·101-s − 215·121-s + 279·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2·4-s + 9/5·5-s + 6/11·11-s + 3·16-s − 2·19-s − 3.59·20-s + 2.23·25-s − 1.09·44-s − 1.48·49-s + 0.981·55-s − 3.37·61-s − 4·64-s + 4·76-s + 27/5·80-s − 3.59·95-s − 4.47·100-s − 2.01·101-s − 1.77·121-s + 2.23·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯

Functional equation

Λ(s)=(731025s/2 ΓC(s)2 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(731025s/2 ΓC(s+1)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 731025731025    =    34⋅52⋅1923^{4} \cdot 5^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 542.753542.753
Root analytic conductor: 4.826704.82670
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 731025, ( :1,1), 1)(4,\ 731025,\ (\ :1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.3461092171.346109217
L(12)L(\frac12) ≈\approx 1.3461092171.346109217
L(2)L(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)
bad3 1 1
5C2C_2 1−9T+p2T2 1 - 9 T + p^{2} T^{2}
19C1C_1 (1+pT)2 ( 1 + p T )^{2}
good2C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
7C2C_2 (1−5T+p2T2)(1+5T+p2T2) ( 1 - 5 T + p^{2} T^{2} )( 1 + 5 T + p^{2} T^{2} )
11C2C_2 (1−3T+p2T2)2 ( 1 - 3 T + p^{2} T^{2} )^{2}
13C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
17C2C_2 (1−15T+p2T2)(1+15T+p2T2) ( 1 - 15 T + p^{2} T^{2} )( 1 + 15 T + p^{2} T^{2} )
23C2C_2 (1−30T+p2T2)(1+30T+p2T2) ( 1 - 30 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} )
29C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
31C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
37C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
41C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
43C2C_2 (1−85T+p2T2)(1+85T+p2T2) ( 1 - 85 T + p^{2} T^{2} )( 1 + 85 T + p^{2} T^{2} )
47C2C_2 (1−75T+p2T2)(1+75T+p2T2) ( 1 - 75 T + p^{2} T^{2} )( 1 + 75 T + p^{2} T^{2} )
53C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
59C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
61C2C_2 (1+103T+p2T2)2 ( 1 + 103 T + p^{2} T^{2} )^{2}
67C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
71C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
73C2C_2 (1−25T+p2T2)(1+25T+p2T2) ( 1 - 25 T + p^{2} T^{2} )( 1 + 25 T + p^{2} T^{2} )
79C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
83C2C_2 (1−90T+p2T2)(1+90T+p2T2) ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} )
89C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
97C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{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

−10.21411103908636647654581436197, −9.550842468510577781871009122963, −9.400871792025640174763422465049, −9.073566620157399150618481857974, −8.614997221610750450508404222912, −8.322030024886761074478638945232, −7.86535701289578131786832250463, −7.15558748746560826001833132181, −6.51995805698792471569870531927, −6.06133435366210021883234367630, −5.97240980051552956224109535402, −5.20346406440331854273489059783, −4.92132482015654401759069009510, −4.37262569318665107873348789605, −4.07599810484994994673126727434, −3.27285391371655388459377380395, −2.74291947162393035842154517126, −1.76823942707055177821230305045, −1.43959609381118701976508448487, −0.40827917194358246516163146790, 0.40827917194358246516163146790, 1.43959609381118701976508448487, 1.76823942707055177821230305045, 2.74291947162393035842154517126, 3.27285391371655388459377380395, 4.07599810484994994673126727434, 4.37262569318665107873348789605, 4.92132482015654401759069009510, 5.20346406440331854273489059783, 5.97240980051552956224109535402, 6.06133435366210021883234367630, 6.51995805698792471569870531927, 7.15558748746560826001833132181, 7.86535701289578131786832250463, 8.322030024886761074478638945232, 8.614997221610750450508404222912, 9.073566620157399150618481857974, 9.400871792025640174763422465049, 9.550842468510577781871009122963, 10.21411103908636647654581436197

Graph of the ZZ-function along the critical line