Properties

Label 4-1450e2-1.1-c1e2-0-22
Degree 44
Conductor 21025002102500
Sign 11
Analytic cond. 134.057134.057
Root an. cond. 3.402693.40269
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 3·4-s − 2·6-s − 5·7-s + 4·8-s − 2·9-s − 2·11-s − 3·12-s − 9·13-s − 10·14-s + 5·16-s − 3·17-s − 4·18-s + 6·19-s + 5·21-s − 4·22-s − 7·23-s − 4·24-s − 18·26-s + 2·27-s − 15·28-s + 2·29-s − 5·31-s + 6·32-s + 2·33-s − 6·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s − 1.88·7-s + 1.41·8-s − 2/3·9-s − 0.603·11-s − 0.866·12-s − 2.49·13-s − 2.67·14-s + 5/4·16-s − 0.727·17-s − 0.942·18-s + 1.37·19-s + 1.09·21-s − 0.852·22-s − 1.45·23-s − 0.816·24-s − 3.53·26-s + 0.384·27-s − 2.83·28-s + 0.371·29-s − 0.898·31-s + 1.06·32-s + 0.348·33-s − 1.02·34-s + ⋯

Functional equation

Λ(s)=(2102500s/2ΓC(s)2L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 2102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2102500s/2ΓC(s+1/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 21025002102500    =    22542922^{2} \cdot 5^{4} \cdot 29^{2}
Sign: 11
Analytic conductor: 134.057134.057
Root analytic conductor: 3.402693.40269
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 2102500, ( :1/2,1/2), 1)(4,\ 2102500,\ (\ :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(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 (1T)2 ( 1 - T )^{2}
5 1 1
29C1C_1 (1T)2 ( 1 - T )^{2}
good3D4D_{4} 1+T+pT2+pT3+p2T4 1 + T + p T^{2} + p T^{3} + p^{2} T^{4}
7D4D_{4} 1+5T+17T2+5pT3+p2T4 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4}
11D4D_{4} 1+2T+10T2+2pT3+p2T4 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4}
13D4D_{4} 1+9T+43T2+9pT3+p2T4 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+3T+7T2+3pT3+p2T4 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4}
19D4D_{4} 16T+34T26pT3+p2T4 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+7T+55T2+7pT3+p2T4 1 + 7 T + 55 T^{2} + 7 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+5T+39T2+5pT3+p2T4 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4}
37D4D_{4} 12T42T22pT3+p2T4 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+10T+94T2+10pT3+p2T4 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+3T+85T2+3pT3+p2T4 1 + 3 T + 85 T^{2} + 3 p T^{3} + p^{2} T^{4}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53D4D_{4} 1+3T+79T2+3pT3+p2T4 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4}
59D4D_{4} 19T+109T29pT3+p2T4 1 - 9 T + 109 T^{2} - 9 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+17T+165T2+17pT3+p2T4 1 + 17 T + 165 T^{2} + 17 p T^{3} + p^{2} T^{4}
67D4D_{4} 14T+86T24pT3+p2T4 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73D4D_{4} 113T+107T213pT3+p2T4 1 - 13 T + 107 T^{2} - 13 p T^{3} + p^{2} T^{4}
79D4D_{4} 1T+129T2pT3+p2T4 1 - T + 129 T^{2} - p T^{3} + p^{2} T^{4}
83D4D_{4} 110T+178T210pT3+p2T4 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+22T+286T2+22pT3+p2T4 1 + 22 T + 286 T^{2} + 22 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+13T+207T2+13pT3+p2T4 1 + 13 T + 207 T^{2} + 13 p T^{3} + p^{2} T^{4}
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   L(s)=p j=14(1αj,pps)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

−9.407990081614753518438997617253, −9.349563612736162688935347605120, −8.157589861722829361886774702904, −8.122977075012076541251019641000, −7.52883759681880672876433442106, −6.99003145307253373700181367280, −6.70473469805233267182206547619, −6.56078543410263305001936796313, −5.84236168494095225366956922217, −5.49718281366288437666415243477, −5.22680105345781307133436841878, −4.87870545989729162912470542411, −4.18060447012509745400533690702, −3.78509387676597826168849823887, −3.12032659178808281312473073020, −2.77834456180461400016046101889, −2.50474533305269970826657323526, −1.69465286739611247746851171454, 0, 0, 1.69465286739611247746851171454, 2.50474533305269970826657323526, 2.77834456180461400016046101889, 3.12032659178808281312473073020, 3.78509387676597826168849823887, 4.18060447012509745400533690702, 4.87870545989729162912470542411, 5.22680105345781307133436841878, 5.49718281366288437666415243477, 5.84236168494095225366956922217, 6.56078543410263305001936796313, 6.70473469805233267182206547619, 6.99003145307253373700181367280, 7.52883759681880672876433442106, 8.122977075012076541251019641000, 8.157589861722829361886774702904, 9.349563612736162688935347605120, 9.407990081614753518438997617253

Graph of the ZZ-function along the critical line