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Properties

Label 4-75411-1.1-c1e2-0-5
Degree 44
Conductor 7541175411
Sign −1-1
Analytic cond. 4.808274.80827
Root an. cond. 1.480801.48080
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  + 4-s − 2·7-s − 6·13-s − 3·16-s − 19-s − 4·25-s − 2·28-s − 6·31-s − 4·37-s − 4·43-s − 3·49-s − 6·52-s + 12·61-s − 7·64-s − 2·67-s − 76-s + 4·79-s + 12·91-s + 18·97-s − 4·100-s − 24·103-s + 14·109-s + 6·112-s − 14·121-s − 6·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.755·7-s − 1.66·13-s − 3/4·16-s − 0.229·19-s − 4/5·25-s − 0.377·28-s − 1.07·31-s − 0.657·37-s − 0.609·43-s − 3/7·49-s − 0.832·52-s + 1.53·61-s − 7/8·64-s − 0.244·67-s − 0.114·76-s + 0.450·79-s + 1.25·91-s + 1.82·97-s − 2/5·100-s − 2.36·103-s + 1.34·109-s + 0.566·112-s − 1.27·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(75411s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 75411 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(75411s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 75411 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 7541175411    =    34⋅72⋅193^{4} \cdot 7^{2} \cdot 19
Sign: −1-1
Analytic conductor: 4.808274.80827
Root analytic conductor: 1.480801.48080
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 75411, ( :1/2,1/2), −1)(4,\ 75411,\ (\ :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)
bad3 1 1
7C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
19C1C_1×\timesC2C_2 (1−T)(1+2T+pT2) ( 1 - T )( 1 + 2 T + p T^{2} )
good2C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
5C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
11C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1+2T+pT2)(1+4T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
23C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
29C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1−2T+pT2)(1+8T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
41C22C_2^2 1+22T2+p2T4 1 + 22 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+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
53C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
59C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−10T+pT2)(1−2T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−2T+pT2)(1+4T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
73C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−16T+pT2)(1−2T+pT2) ( 1 - 16 T + p T^{2} )( 1 - 2 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

−9.451808115510595304934440107969, −9.284493818243571866566646306503, −8.523866780545975415324122899128, −7.962936558854149882361927378320, −7.33995880204398288349734876353, −7.00108325834014391470111729249, −6.56218917408877805692663255116, −5.91990419890763971288746040554, −5.26463306290841073268659490826, −4.75590964208523306519663302548, −3.97450961899134079626108319531, −3.28792973835471695930734978908, −2.48940279941837917395101269178, −1.91806497648452369136382678113, 0, 1.91806497648452369136382678113, 2.48940279941837917395101269178, 3.28792973835471695930734978908, 3.97450961899134079626108319531, 4.75590964208523306519663302548, 5.26463306290841073268659490826, 5.91990419890763971288746040554, 6.56218917408877805692663255116, 7.00108325834014391470111729249, 7.33995880204398288349734876353, 7.962936558854149882361927378320, 8.523866780545975415324122899128, 9.284493818243571866566646306503, 9.451808115510595304934440107969

Graph of the ZZ-function along the critical line