Properties

Label 4-2352e2-1.1-c0e2-0-3
Degree 44
Conductor 55319045531904
Sign 11
Analytic cond. 1.377801.37780
Root an. cond. 1.083421.08342
Motivic weight 00
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  + 3-s − 2·19-s + 25-s − 27-s + 2·31-s + 2·37-s − 2·57-s + 75-s − 81-s + 2·93-s + 2·103-s − 2·109-s + 2·111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 3-s − 2·19-s + 25-s − 27-s + 2·31-s + 2·37-s − 2·57-s + 75-s − 81-s + 2·93-s + 2·103-s − 2·109-s + 2·111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

Λ(s)=(5531904s/2ΓC(s)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(5531904s/2ΓC(s)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 55319045531904    =    2832742^{8} \cdot 3^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 1.377801.37780
Root analytic conductor: 1.083421.08342
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 5531904, ( :0,0), 1)(4,\ 5531904,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.6827317611.682731761
L(12)L(\frac12) \approx 1.6827317611.682731761
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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 1T+T2 1 - T + T^{2}
7 1 1
good5C22C_2^2 1T2+T4 1 - T^{2} + T^{4}
11C22C_2^2 1T2+T4 1 - T^{2} + T^{4}
13C1C_1×\timesC1C_1 (1T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
17C22C_2^2 1T2+T4 1 - T^{2} + T^{4}
19C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
23C22C_2^2 1T2+T4 1 - T^{2} + T^{4}
29C1C_1×\timesC1C_1 (1T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
31C2C_2 (1T+T2)2 ( 1 - T + T^{2} )^{2}
37C2C_2 (1T+T2)2 ( 1 - T + T^{2} )^{2}
41C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
43C1C_1×\timesC1C_1 (1T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
47C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
53C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
59C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
61C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
67C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
71C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
73C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
79C2C_2 (1T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
83C1C_1×\timesC1C_1 (1T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
89C22C_2^2 1T2+T4 1 - T^{2} + T^{4}
97C1C_1×\timesC1C_1 (1T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
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   L(s)=p j=14(1αj,pps)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

−9.347913553467959116379971356881, −8.829984375487463093238543542460, −8.620886641361845600039444231418, −8.191822468902443548097144435338, −8.059504669325826181063202993505, −7.51604790775848753725465989904, −7.18155410114895145859927380363, −6.51005196797831603293376521922, −6.28431738649007638057939817588, −6.12154568390454536581049972812, −5.37298834629383443464695512859, −4.92489926098771545132311834879, −4.44880349665391334457367252945, −4.12244356078066379432436476872, −3.73112614678026613213519520011, −2.85487618404965542030039888056, −2.84971420831768090795136390045, −2.29856121296980983520458619846, −1.71918953262364424806938255793, −0.845557552588317244927063311378, 0.845557552588317244927063311378, 1.71918953262364424806938255793, 2.29856121296980983520458619846, 2.84971420831768090795136390045, 2.85487618404965542030039888056, 3.73112614678026613213519520011, 4.12244356078066379432436476872, 4.44880349665391334457367252945, 4.92489926098771545132311834879, 5.37298834629383443464695512859, 6.12154568390454536581049972812, 6.28431738649007638057939817588, 6.51005196797831603293376521922, 7.18155410114895145859927380363, 7.51604790775848753725465989904, 8.059504669325826181063202993505, 8.191822468902443548097144435338, 8.620886641361845600039444231418, 8.829984375487463093238543542460, 9.347913553467959116379971356881

Graph of the ZZ-function along the critical line