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Properties

Label 4-2163200-1.1-c1e2-0-1
Degree 44
Conductor 21632002163200
Sign 11
Analytic cond. 137.927137.927
Root an. cond. 3.426983.42698
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 6·9-s + 4·11-s + 4·17-s + 4·19-s + 25-s − 4·27-s + 16·33-s + 4·41-s + 12·43-s − 14·49-s + 16·51-s + 16·57-s + 12·59-s − 4·73-s + 4·75-s − 37·81-s + 24·83-s − 12·89-s + 4·97-s + 24·99-s − 4·107-s + 4·113-s − 10·121-s + 16·123-s + 127-s + 48·129-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s + 1.20·11-s + 0.970·17-s + 0.917·19-s + 1/5·25-s − 0.769·27-s + 2.78·33-s + 0.624·41-s + 1.82·43-s − 2·49-s + 2.24·51-s + 2.11·57-s + 1.56·59-s − 0.468·73-s + 0.461·75-s − 4.11·81-s + 2.63·83-s − 1.27·89-s + 0.406·97-s + 2.41·99-s − 0.386·107-s + 0.376·113-s − 0.909·121-s + 1.44·123-s + 0.0887·127-s + 4.22·129-s + ⋯

Functional equation

Λ(s)=(2163200s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2163200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2163200s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2163200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 21632002163200    =    29⋅52⋅1322^{9} \cdot 5^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 137.927137.927
Root analytic conductor: 3.426983.42698
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2163200, ( :1/2,1/2), 1)(4,\ 2163200,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.2238741086.223874108
L(12)L(\frac12) ≈\approx 6.2238741086.223874108
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
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
13C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
23C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
43C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 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 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
61C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
67C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
71C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
73C2C_2 (1+2T+pT2)2 ( 1 + 2 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)2 ( 1 - 12 T + p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{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

−7.76319041513224922753287966748, −7.44095337164983387150060505996, −7.22602053082602449735325415510, −6.33588991532526860429499225622, −6.28727870819603758957301866812, −5.43479364046733402754835461064, −5.24172374409154567149889570954, −4.38796175153128331239805240266, −3.83609522350488929158971696987, −3.73594560911583979412114290925, −2.95784688889706959154847700703, −2.93871105322865322021254198939, −2.18201972761241363090254415749, −1.63546887865200897104772810976, −0.910977372178463170757222920731, 0.910977372178463170757222920731, 1.63546887865200897104772810976, 2.18201972761241363090254415749, 2.93871105322865322021254198939, 2.95784688889706959154847700703, 3.73594560911583979412114290925, 3.83609522350488929158971696987, 4.38796175153128331239805240266, 5.24172374409154567149889570954, 5.43479364046733402754835461064, 6.28727870819603758957301866812, 6.33588991532526860429499225622, 7.22602053082602449735325415510, 7.44095337164983387150060505996, 7.76319041513224922753287966748

Graph of the ZZ-function along the critical line