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Properties

Label 4-700e2-1.1-c2e2-0-0
Degree 44
Conductor 490000490000
Sign 11
Analytic cond. 363.802363.802
Root an. cond. 4.367334.36733
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  − 3·3-s + 14·7-s − 3·9-s − 15·11-s − 51·17-s + 27·19-s − 42·21-s − 9·23-s + 18·27-s − 12·29-s − 21·31-s + 45·33-s + 31·37-s − 20·43-s − 75·47-s + 147·49-s + 153·51-s − 57·53-s − 81·57-s − 141·59-s − 141·61-s − 42·63-s − 49·67-s + 27·69-s − 252·71-s + 45·73-s − 210·77-s + ⋯
L(s)  = 1  − 3-s + 2·7-s − 1/3·9-s − 1.36·11-s − 3·17-s + 1.42·19-s − 2·21-s − 0.391·23-s + 2/3·27-s − 0.413·29-s − 0.677·31-s + 1.36·33-s + 0.837·37-s − 0.465·43-s − 1.59·47-s + 3·49-s + 3·51-s − 1.07·53-s − 1.42·57-s − 2.38·59-s − 2.31·61-s − 2/3·63-s − 0.731·67-s + 9/23·69-s − 3.54·71-s + 0.616·73-s − 2.72·77-s + ⋯

Functional equation

Λ(s)=(490000s/2 ΓC(s)2 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(490000s/2 ΓC(s+1)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 490000490000    =    24⋅54⋅722^{4} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 363.802363.802
Root analytic conductor: 4.367334.36733
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 490000, ( :1,1), 1)(4,\ 490000,\ (\ :1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.28560405910.2856040591
L(12)L(\frac12) ≈\approx 0.28560405910.2856040591
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)
bad2 1 1
5 1 1
7C1C_1 (1−pT)2 ( 1 - p T )^{2}
good3C22C_2^2 1+pT+4pT2+p3T3+p4T4 1 + p T + 4 p T^{2} + p^{3} T^{3} + p^{4} T^{4}
11C22C_2^2 1+15T+104T2+15p2T3+p4T4 1 + 15 T + 104 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4}
13C2C_2 (1−22T+p2T2)(1+22T+p2T2) ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} )
17C1C_1×\timesC2C_2 (1+pT)2(1+pT+p2T2) ( 1 + p T )^{2}( 1 + p T + p^{2} T^{2} )
19C22C_2^2 1−27T+604T2−27p2T3+p4T4 1 - 27 T + 604 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4}
23C22C_2^2 1+9T−448T2+9p2T3+p4T4 1 + 9 T - 448 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4}
29C2C_2 (1+6T+p2T2)2 ( 1 + 6 T + p^{2} T^{2} )^{2}
31C22C_2^2 1+21T+1108T2+21p2T3+p4T4 1 + 21 T + 1108 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4}
37C22C_2^2 1−31T−408T2−31p2T3+p4T4 1 - 31 T - 408 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4}
41C22C_2^2 1−290T2+p4T4 1 - 290 T^{2} + p^{4} T^{4}
43C2C_2 (1+10T+p2T2)2 ( 1 + 10 T + p^{2} T^{2} )^{2}
47C22C_2^2 1+75T+4084T2+75p2T3+p4T4 1 + 75 T + 4084 T^{2} + 75 p^{2} T^{3} + p^{4} T^{4}
53C22C_2^2 1+57T+440T2+57p2T3+p4T4 1 + 57 T + 440 T^{2} + 57 p^{2} T^{3} + p^{4} T^{4}
59C22C_2^2 1+141T+10108T2+141p2T3+p4T4 1 + 141 T + 10108 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4}
61C22C_2^2 1+141T+10348T2+141p2T3+p4T4 1 + 141 T + 10348 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4}
67C22C_2^2 1+49T−2088T2+49p2T3+p4T4 1 + 49 T - 2088 T^{2} + 49 p^{2} T^{3} + p^{4} T^{4}
71C2C_2 (1+126T+p2T2)2 ( 1 + 126 T + p^{2} T^{2} )^{2}
73C22C_2^2 1−45T+6004T2−45p2T3+p4T4 1 - 45 T + 6004 T^{2} - 45 p^{2} T^{3} + p^{4} T^{4}
79C22C_2^2 1−73T−912T2−73p2T3+p4T4 1 - 73 T - 912 T^{2} - 73 p^{2} T^{3} + p^{4} T^{4}
83C22C_2^2 1−13586T2+p4T4 1 - 13586 T^{2} + p^{4} T^{4}
89C22C_2^2 1−99T+11188T2−99p2T3+p4T4 1 - 99 T + 11188 T^{2} - 99 p^{2} T^{3} + p^{4} T^{4}
97C22C_2^2 1−18050T2+p4T4 1 - 18050 T^{2} + p^{4} T^{4}
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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

−10.77203314139928659560084928212, −10.30889164606294289802650981018, −9.551712626442508563308807321574, −8.969251897109009723270315925100, −8.905804246121212237653338476789, −8.184481127579734496679130315611, −7.78459397375095089021536461444, −7.56310690112796074825511529298, −7.03217599242051044928547920826, −6.21674851687892022468422661497, −6.02302629311301303435145859666, −5.46576852031566262897330453106, −4.91050384184050559143940697438, −4.60002160232112338680603805334, −4.47443511609953302678460723697, −3.29324419362551245130096556408, −2.69690188492565920934702722165, −1.94570219692609133228107018188, −1.52144631369747675148343201508, −0.19625571118695790255210641001, 0.19625571118695790255210641001, 1.52144631369747675148343201508, 1.94570219692609133228107018188, 2.69690188492565920934702722165, 3.29324419362551245130096556408, 4.47443511609953302678460723697, 4.60002160232112338680603805334, 4.91050384184050559143940697438, 5.46576852031566262897330453106, 6.02302629311301303435145859666, 6.21674851687892022468422661497, 7.03217599242051044928547920826, 7.56310690112796074825511529298, 7.78459397375095089021536461444, 8.184481127579734496679130315611, 8.905804246121212237653338476789, 8.969251897109009723270315925100, 9.551712626442508563308807321574, 10.30889164606294289802650981018, 10.77203314139928659560084928212

Graph of the ZZ-function along the critical line