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Properties

Label 4-3960e2-1.1-c1e2-0-2
Degree 44
Conductor 1568160015681600
Sign 11
Analytic cond. 999.872999.872
Root an. cond. 5.623235.62323
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·7-s − 2·11-s + 2·13-s + 2·17-s + 8·23-s + 3·25-s − 4·29-s − 4·35-s + 12·37-s − 12·41-s + 2·43-s − 8·47-s + 6·49-s − 4·55-s − 8·59-s + 20·61-s + 4·65-s + 4·67-s − 4·71-s + 18·73-s + 4·77-s − 12·79-s + 2·83-s + 4·85-s + 16·89-s − 4·91-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s − 0.603·11-s + 0.554·13-s + 0.485·17-s + 1.66·23-s + 3/5·25-s − 0.742·29-s − 0.676·35-s + 1.97·37-s − 1.87·41-s + 0.304·43-s − 1.16·47-s + 6/7·49-s − 0.539·55-s − 1.04·59-s + 2.56·61-s + 0.496·65-s + 0.488·67-s − 0.474·71-s + 2.10·73-s + 0.455·77-s − 1.35·79-s + 0.219·83-s + 0.433·85-s + 1.69·89-s − 0.419·91-s + ⋯

Functional equation

Λ(s)=(15681600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(15681600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1568160015681600    =    26⋅34⋅52⋅1122^{6} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 999.872999.872
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 15681600, ( :1/2,1/2), 1)(4,\ 15681600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0904711883.090471188
L(12)L(\frac12) ≈\approx 3.0904711883.090471188
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)
bad2 1 1
3 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
11C1C_1 (1+T)2 ( 1 + T )^{2}
good7D4D_{4} 1+2T−2T2+2pT3+p2T4 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
13C4C_4 1−2T+10T2−2pT3+p2T4 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−2T+18T2−2pT3+p2T4 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4}
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43D4D_{4} 1−2T+70T2−2pT3+p2T4 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4}
47C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
53C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
59C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
61C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
67D4D_{4} 1−4T+70T2−4pT3+p2T4 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+4T+78T2+4pT3+p2T4 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−18T+210T2−18pT3+p2T4 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+12T+126T2+12pT3+p2T4 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−2T+14T2−2pT3+p2T4 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−16T+174T2−16pT3+p2T4 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−8T+142T2−8pT3+p2T4 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} 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

−8.681927373356142504717409803471, −8.237266220587868450622337279627, −8.008151874342029604760356293887, −7.52444327848299020739194379237, −6.99534207064231104535886260551, −6.90183151129377935470332399835, −6.33194746375446376072284713970, −6.16702117471608079278945832342, −5.59790009145838515672466575544, −5.41082390874012797406560280836, −4.86482019649560809189618081266, −4.73108754265234899709649356983, −3.81727508144634279978041275167, −3.71360024776516049312484128673, −2.99048750587129398976343581364, −2.89622172001196878056084332535, −2.22995185898318267104759345975, −1.78009553567555265826743142776, −1.07440916665369499545697458968, −0.56887589256857787452940651825, 0.56887589256857787452940651825, 1.07440916665369499545697458968, 1.78009553567555265826743142776, 2.22995185898318267104759345975, 2.89622172001196878056084332535, 2.99048750587129398976343581364, 3.71360024776516049312484128673, 3.81727508144634279978041275167, 4.73108754265234899709649356983, 4.86482019649560809189618081266, 5.41082390874012797406560280836, 5.59790009145838515672466575544, 6.16702117471608079278945832342, 6.33194746375446376072284713970, 6.90183151129377935470332399835, 6.99534207064231104535886260551, 7.52444327848299020739194379237, 8.008151874342029604760356293887, 8.237266220587868450622337279627, 8.681927373356142504717409803471

Graph of the ZZ-function along the critical line