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Properties

Label 4-855e2-1.1-c1e2-0-5
Degree 44
Conductor 731025731025
Sign 11
Analytic cond. 46.610746.6107
Root an. cond. 2.612892.61289
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 4·7-s − 3·16-s + 2·19-s − 25-s + 4·28-s + 8·31-s + 8·37-s − 4·43-s + 2·49-s − 12·61-s − 7·64-s − 8·67-s + 12·73-s + 2·76-s + 8·79-s + 8·97-s − 100-s + 24·103-s + 12·109-s − 12·112-s − 2·121-s + 8·124-s + 127-s + 131-s + 8·133-s + 137-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.51·7-s − 3/4·16-s + 0.458·19-s − 1/5·25-s + 0.755·28-s + 1.43·31-s + 1.31·37-s − 0.609·43-s + 2/7·49-s − 1.53·61-s − 7/8·64-s − 0.977·67-s + 1.40·73-s + 0.229·76-s + 0.900·79-s + 0.812·97-s − 0.0999·100-s + 2.36·103-s + 1.14·109-s − 1.13·112-s − 0.181·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-s + 0.0854·137-s + ⋯

Functional equation

Λ(s)=(731025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(731025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{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: 46.610746.6107
Root analytic conductor: 2.612892.61289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 731025, ( :1/2,1/2), 1)(4,\ 731025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8216518812.821651881
L(12)L(\frac12) ≈\approx 2.8216518812.821651881
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)
bad3 1 1
5C2C_2 1+T2 1 + T^{2}
19C1C_1 (1−T)2 ( 1 - T )^{2}
good2C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
7C2C_2×\timesC2C_2 (1−4T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
11C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
37C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
41C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C22C_2^2 1−74T2+p2T4 1 - 74 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−2T+pT2)(1+14T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−4T+pT2)(1+12T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
73C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
79C2C_2×\timesC2C_2 (1−12T+pT2)(1+4T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
89C22C_2^2 1+102T2+p2T4 1 + 102 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−14T+pT2)(1+6T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 6 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

−8.167798775468386922497017119471, −7.81629925534961684383397522072, −7.56286056811260075809566840048, −7.01862500787748876169081978041, −6.39492674973921678255633336802, −6.19358130523662731614596938467, −5.54933623644349374683330696508, −4.92586372856273525185184956084, −4.62565435014953298320952525244, −4.29428437177532962119033259859, −3.40322077346350739176953316718, −2.88611679888280351701569505957, −2.15652647337383028968047721099, −1.72158885252222435895990718582, −0.853110982786161162683313101739, 0.853110982786161162683313101739, 1.72158885252222435895990718582, 2.15652647337383028968047721099, 2.88611679888280351701569505957, 3.40322077346350739176953316718, 4.29428437177532962119033259859, 4.62565435014953298320952525244, 4.92586372856273525185184956084, 5.54933623644349374683330696508, 6.19358130523662731614596938467, 6.39492674973921678255633336802, 7.01862500787748876169081978041, 7.56286056811260075809566840048, 7.81629925534961684383397522072, 8.167798775468386922497017119471

Graph of the ZZ-function along the critical line