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Properties

Label 4-31212-1.1-c1e2-0-3
Degree 44
Conductor 3121231212
Sign −1-1
Analytic cond. 1.990101.99010
Root an. cond. 1.187731.18773
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 4·7-s + 9-s − 12-s − 12·13-s + 16-s + 8·19-s + 4·21-s + 6·25-s − 27-s − 4·28-s − 12·31-s + 36-s − 8·37-s + 12·39-s − 8·43-s − 48-s − 2·49-s − 12·52-s − 8·57-s − 8·61-s − 4·63-s + 64-s − 24·67-s + 4·73-s − 6·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 1.51·7-s + 1/3·9-s − 0.288·12-s − 3.32·13-s + 1/4·16-s + 1.83·19-s + 0.872·21-s + 6/5·25-s − 0.192·27-s − 0.755·28-s − 2.15·31-s + 1/6·36-s − 1.31·37-s + 1.92·39-s − 1.21·43-s − 0.144·48-s − 2/7·49-s − 1.66·52-s − 1.05·57-s − 1.02·61-s − 0.503·63-s + 1/8·64-s − 2.93·67-s + 0.468·73-s − 0.692·75-s + ⋯

Functional equation

Λ(s)=(31212s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 31212 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(31212s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 31212 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3121231212    =    22⋅33⋅1722^{2} \cdot 3^{3} \cdot 17^{2}
Sign: −1-1
Analytic conductor: 1.990101.99010
Root analytic conductor: 1.187731.18773
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 31212, ( :1/2,1/2), −1)(4,\ 31212,\ (\ :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(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
3C1C_1 1+T 1 + T
17C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
7C2C_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+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{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 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
31C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
37C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
53C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
67C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
71C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
73C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
79C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
97C2C_2 (1−6T+pT2)2 ( 1 - 6 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

−10.13636675520507553497406453887, −9.809680223299837765262582433989, −9.330186221363971309576111465163, −8.951168622566808298087877447389, −7.58460004774890445683046626123, −7.50224155158573118874050462938, −6.99640780173569532110131873676, −6.52498883063792512928187502617, −5.71929507199710618712172159386, −5.00716852164944696644803170219, −4.84625070630626219217033988858, −3.34589616522309311397390467846, −3.08824258991321588799800323044, −2.01050304782327789083256558357, 0, 2.01050304782327789083256558357, 3.08824258991321588799800323044, 3.34589616522309311397390467846, 4.84625070630626219217033988858, 5.00716852164944696644803170219, 5.71929507199710618712172159386, 6.52498883063792512928187502617, 6.99640780173569532110131873676, 7.50224155158573118874050462938, 7.58460004774890445683046626123, 8.951168622566808298087877447389, 9.330186221363971309576111465163, 9.809680223299837765262582433989, 10.13636675520507553497406453887

Graph of the ZZ-function along the critical line