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Properties

Label 4-249696-1.1-c1e2-0-1
Degree 44
Conductor 249696249696
Sign 11
Analytic cond. 15.920815.9208
Root an. cond. 1.997521.99752
Motivic weight 11
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  + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s + 4·13-s + 16-s + 18-s + 12·23-s − 24-s − 10·25-s + 4·26-s − 27-s + 32-s + 36-s + 16·37-s − 4·39-s + 12·46-s − 24·47-s − 48-s − 10·49-s − 10·50-s + 4·52-s − 54-s + 24·59-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.235·18-s + 2.50·23-s − 0.204·24-s − 2·25-s + 0.784·26-s − 0.192·27-s + 0.176·32-s + 1/6·36-s + 2.63·37-s − 0.640·39-s + 1.76·46-s − 3.50·47-s − 0.144·48-s − 1.42·49-s − 1.41·50-s + 0.554·52-s − 0.136·54-s + 3.12·59-s + ⋯

Functional equation

Λ(s)=(249696s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 249696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(249696s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 249696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 249696249696    =    25⋅33⋅1722^{5} \cdot 3^{3} \cdot 17^{2}
Sign: 11
Analytic conductor: 15.920815.9208
Root analytic conductor: 1.997521.99752
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 249696, ( :1/2,1/2), 1)(4,\ 249696,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4698409612.469840961
L(12)L(\frac12) ≈\approx 2.4698409612.469840961
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 1 - T
3C1C_1 1+T 1 + T
17C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
73C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
79C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
83C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
89C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} )
97C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{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

−8.745924693621362183100177299272, −8.584868551115419667042947592500, −7.81014283472581039979750102513, −7.57323216500839527256914149425, −6.70932306148654350305246926067, −6.62762171992257665130856519803, −5.95967651264422152670977788910, −5.60715979666329449780909401068, −4.94376122657450226002634205632, −4.60759445774375135200288872743, −3.84609690192901140186989283204, −3.42809860399723990451499242436, −2.71372337045158087575655535987, −1.81193945201237649695550582777, −0.943136411131521382953657369703, 0.943136411131521382953657369703, 1.81193945201237649695550582777, 2.71372337045158087575655535987, 3.42809860399723990451499242436, 3.84609690192901140186989283204, 4.60759445774375135200288872743, 4.94376122657450226002634205632, 5.60715979666329449780909401068, 5.95967651264422152670977788910, 6.62762171992257665130856519803, 6.70932306148654350305246926067, 7.57323216500839527256914149425, 7.81014283472581039979750102513, 8.584868551115419667042947592500, 8.745924693621362183100177299272

Graph of the ZZ-function along the critical line