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Properties

Label 4-857500-1.1-c1e2-0-5
Degree 44
Conductor 857500857500
Sign −1-1
Analytic cond. 54.674954.6749
Root an. cond. 2.719232.71923
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 7-s − 4·8-s − 5·9-s + 6·11-s + 2·14-s + 5·16-s + 10·18-s − 12·22-s − 3·28-s − 12·29-s − 6·32-s − 15·36-s − 16·37-s − 16·43-s + 18·44-s + 49-s + 24·53-s + 4·56-s + 24·58-s + 5·63-s + 7·64-s + 14·67-s + 12·71-s + 20·72-s + 32·74-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s − 5/3·9-s + 1.80·11-s + 0.534·14-s + 5/4·16-s + 2.35·18-s − 2.55·22-s − 0.566·28-s − 2.22·29-s − 1.06·32-s − 5/2·36-s − 2.63·37-s − 2.43·43-s + 2.71·44-s + 1/7·49-s + 3.29·53-s + 0.534·56-s + 3.15·58-s + 0.629·63-s + 7/8·64-s + 1.71·67-s + 1.42·71-s + 2.35·72-s + 3.71·74-s + ⋯

Functional equation

Λ(s)=(857500s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(857500s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 857500857500    =    22⋅54⋅732^{2} \cdot 5^{4} \cdot 7^{3}
Sign: −1-1
Analytic conductor: 54.674954.6749
Root analytic conductor: 2.719232.71923
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 857500, ( :1/2,1/2), −1)(4,\ 857500,\ (\ :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)
bad2C1C_1 (1+T)2 ( 1 + T )^{2}
5 1 1
7C1C_1 1+T 1 + T
good3C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
19C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
41C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
43C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
47C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
53C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
79C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
83C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
89C2C_2 (1−15T+pT2)(1+15T+pT2) ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} )
97C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 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

−8.241263185170336936107387248232, −7.62583420866028467275275698765, −7.02864082573576697328686349834, −6.80760429487312719242044771491, −6.40152272675796392235796567725, −5.91589072886351065562126787974, −5.26770803610804114861303773339, −5.17711518617506807839806183328, −3.76060769659553506276694871497, −3.69265834103033057215511968857, −3.22985803075868355514464176224, −2.12269688406492296074462119899, −2.02089552612607080439032267478, −0.926909876778585217976162918755, 0, 0.926909876778585217976162918755, 2.02089552612607080439032267478, 2.12269688406492296074462119899, 3.22985803075868355514464176224, 3.69265834103033057215511968857, 3.76060769659553506276694871497, 5.17711518617506807839806183328, 5.26770803610804114861303773339, 5.91589072886351065562126787974, 6.40152272675796392235796567725, 6.80760429487312719242044771491, 7.02864082573576697328686349834, 7.62583420866028467275275698765, 8.241263185170336936107387248232

Graph of the ZZ-function along the critical line