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Properties

Label 4-936e2-1.1-c1e2-0-0
Degree 44
Conductor 876096876096
Sign 11
Analytic cond. 55.860655.8606
Root an. cond. 2.733862.73386
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  − 8·5-s − 2·13-s + 38·25-s + 16·29-s + 12·37-s − 24·41-s + 2·49-s − 4·61-s + 16·65-s − 20·73-s + 24·89-s + 28·97-s − 16·101-s − 28·109-s + 16·113-s − 6·121-s − 136·125-s + 127-s + 131-s + 137-s + 139-s − 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3.57·5-s − 0.554·13-s + 38/5·25-s + 2.97·29-s + 1.97·37-s − 3.74·41-s + 2/7·49-s − 0.512·61-s + 1.98·65-s − 2.34·73-s + 2.54·89-s + 2.84·97-s − 1.59·101-s − 2.68·109-s + 1.50·113-s − 0.545·121-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=(876096s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(876096s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 876096876096    =    26⋅34⋅1322^{6} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 55.860655.8606
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 876096, ( :1/2,1/2), 1)(4,\ 876096,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.49653902590.4965390259
L(12)L(\frac12) ≈\approx 0.49653902590.4965390259
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
13C1C_1 (1+T)2 ( 1 + T )^{2}
good5C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
41C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
43C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
73C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
79C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{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

−8.201733704151596185987903064087, −7.79060854775042365605164468705, −7.47869322356582658895088316447, −6.94787299886291680521490460261, −6.67145474632920446177992614245, −6.19578640602053014565524878691, −5.05645028216135290095057077475, −4.88053255158482796760819813001, −4.39469345798104888310840654068, −4.10665661851761688347162074488, −3.38922081885229144696621278738, −3.17258330639560750206393636102, −2.58928593205915212490307519815, −1.17283728352626636257793590231, −0.38897534472433013470827993329, 0.38897534472433013470827993329, 1.17283728352626636257793590231, 2.58928593205915212490307519815, 3.17258330639560750206393636102, 3.38922081885229144696621278738, 4.10665661851761688347162074488, 4.39469345798104888310840654068, 4.88053255158482796760819813001, 5.05645028216135290095057077475, 6.19578640602053014565524878691, 6.67145474632920446177992614245, 6.94787299886291680521490460261, 7.47869322356582658895088316447, 7.79060854775042365605164468705, 8.201733704151596185987903064087

Graph of the ZZ-function along the critical line