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Properties

Label 4-240e2-1.1-c1e2-0-34
Degree 44
Conductor 5760057600
Sign −1-1
Analytic cond. 3.672623.67262
Root an. cond. 1.384341.38434
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s + 9-s − 4·17-s − 12·23-s + 25-s + 4·41-s − 4·47-s + 2·49-s − 4·63-s − 8·71-s − 12·73-s − 8·79-s + 81-s − 12·89-s − 12·97-s + 4·103-s + 12·113-s + 16·119-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + ⋯
L(s)  = 1  − 1.51·7-s + 1/3·9-s − 0.970·17-s − 2.50·23-s + 1/5·25-s + 0.624·41-s − 0.583·47-s + 2/7·49-s − 0.503·63-s − 0.949·71-s − 1.40·73-s − 0.900·79-s + 1/9·81-s − 1.27·89-s − 1.21·97-s + 0.394·103-s + 1.12·113-s + 1.46·119-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.323·153-s + 0.0798·157-s + ⋯

Functional equation

Λ(s)=(57600s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(57600s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5760057600    =    28⋅32⋅522^{8} \cdot 3^{2} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 3.672623.67262
Root analytic conductor: 1.384341.38434
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 57600, ( :1/2,1/2), −1)(4,\ 57600,\ (\ :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)
bad2 1 1
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good7C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1+4T+pT2)(1+8T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
41C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
43C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
53C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
59C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
61C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
67C22C_2^2 1+86T2+p2T4 1 + 86 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1−8T+pT2)(1+16T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} )
73C2C_2×\timesC2C_2 (1+2T+pT2)(1+10T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−8T+pT2)(1+16T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} )
83C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2×\timesC2C_2 (1+2T+pT2)(1+10T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{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

−9.789558936963360670762164057052, −9.432331843962428674696661297451, −8.651227420043199698826817811666, −8.349861868559778434364327135898, −7.58552861389566241021822241490, −7.08242127393491403183760755045, −6.54113518303047890819319823827, −6.03142414243910032413180227612, −5.69872404151448590531497577728, −4.58279675140756191403136982971, −4.17610479998371727321127812323, −3.45201131090447634672216722360, −2.73019489142800131650849040974, −1.82414167365812974415618269325, 0, 1.82414167365812974415618269325, 2.73019489142800131650849040974, 3.45201131090447634672216722360, 4.17610479998371727321127812323, 4.58279675140756191403136982971, 5.69872404151448590531497577728, 6.03142414243910032413180227612, 6.54113518303047890819319823827, 7.08242127393491403183760755045, 7.58552861389566241021822241490, 8.349861868559778434364327135898, 8.651227420043199698826817811666, 9.432331843962428674696661297451, 9.789558936963360670762164057052

Graph of the ZZ-function along the critical line