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Properties

Label 4-60e2-1.1-c1e2-0-0
Degree 44
Conductor 36003600
Sign 11
Analytic cond. 0.2295390.229539
Root an. cond. 0.6921720.692172
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  − 2·3-s + 4·7-s + 9-s + 4·13-s − 8·19-s − 8·21-s + 25-s + 4·27-s − 8·31-s + 4·37-s − 8·39-s − 20·43-s − 2·49-s + 16·57-s + 4·61-s + 4·63-s + 4·67-s + 4·73-s − 2·75-s + 16·79-s − 11·81-s + 16·91-s + 16·93-s + 4·97-s + 28·103-s + 4·109-s − 8·111-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.51·7-s + 1/3·9-s + 1.10·13-s − 1.83·19-s − 1.74·21-s + 1/5·25-s + 0.769·27-s − 1.43·31-s + 0.657·37-s − 1.28·39-s − 3.04·43-s − 2/7·49-s + 2.11·57-s + 0.512·61-s + 0.503·63-s + 0.488·67-s + 0.468·73-s − 0.230·75-s + 1.80·79-s − 1.22·81-s + 1.67·91-s + 1.65·93-s + 0.406·97-s + 2.75·103-s + 0.383·109-s − 0.759·111-s + ⋯

Functional equation

Λ(s)=(3600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 36003600    =    24⋅32⋅522^{4} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 0.2295390.229539
Root analytic conductor: 0.6921720.692172
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 3600, ( :1/2,1/2), 1)(4,\ 3600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.61806266770.6180626677
L(12)L(\frac12) ≈\approx 0.61806266770.6180626677
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
3C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good7C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 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+10T+pT2)2 ( 1 + 10 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 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
71C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1−2T+pT2)2 ( 1 - 2 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

−12.64919926214782333798361971460, −11.66835822739294093715813029994, −11.50854453196989171707939391081, −10.83473950301065015207539180976, −10.66869217043212474435745329357, −9.749524176059453166876684221310, −8.552217550781204179646223792184, −8.536040373596851277524077678909, −7.64589778281564359800589548721, −6.57891116465648258947670054106, −6.21444115782927123192608967686, −5.19647848534431193907436500640, −4.78130792717525308450176413839, −3.71119793411816721603957340078, −1.81793015252092636076156145980, 1.81793015252092636076156145980, 3.71119793411816721603957340078, 4.78130792717525308450176413839, 5.19647848534431193907436500640, 6.21444115782927123192608967686, 6.57891116465648258947670054106, 7.64589778281564359800589548721, 8.536040373596851277524077678909, 8.552217550781204179646223792184, 9.749524176059453166876684221310, 10.66869217043212474435745329357, 10.83473950301065015207539180976, 11.50854453196989171707939391081, 11.66835822739294093715813029994, 12.64919926214782333798361971460

Graph of the ZZ-function along the critical line