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Properties

Label 4-1440e2-1.1-c1e2-0-19
Degree 44
Conductor 20736002073600
Sign 11
Analytic cond. 132.214132.214
Root an. cond. 3.390933.39093
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  − 2·5-s − 25-s + 20·29-s + 20·41-s + 14·49-s + 20·61-s − 20·89-s + 4·101-s − 12·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 40·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 0.894·5-s − 1/5·25-s + 3.71·29-s + 3.12·41-s + 2·49-s + 2.56·61-s − 2.11·89-s + 0.398·101-s − 1.14·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

Λ(s)=(2073600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2073600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20736002073600    =    210⋅34⋅522^{10} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 132.214132.214
Root analytic conductor: 3.390933.39093
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2073600, ( :1/2,1/2), 1)(4,\ 2073600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0497843792.049784379
L(12)L(\frac12) ≈\approx 2.0497843792.049784379
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
5C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good7C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
29C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
67C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
89C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 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

−9.654155887792734734389719725953, −9.420844271631511952858485504146, −8.724196582595219545465983625642, −8.534964065312342776141551775522, −8.162169145789519053179197596571, −7.79941900119667763682971898495, −7.22451300132872997519292084108, −7.07682236961043110027976443929, −6.34110085891932730965192004051, −6.26634789343380574013748111604, −5.45167070235756735737882026857, −5.27114339318429747720415150274, −4.40214920062796692947090991854, −4.27324654952734499041190090775, −3.93107504365283413400261443332, −3.11797907156419186966667701453, −2.65571018849977082810313264801, −2.30702506626782030106039441031, −1.10903140412871219835651823945, −0.70081637769018333116178111448, 0.70081637769018333116178111448, 1.10903140412871219835651823945, 2.30702506626782030106039441031, 2.65571018849977082810313264801, 3.11797907156419186966667701453, 3.93107504365283413400261443332, 4.27324654952734499041190090775, 4.40214920062796692947090991854, 5.27114339318429747720415150274, 5.45167070235756735737882026857, 6.26634789343380574013748111604, 6.34110085891932730965192004051, 7.07682236961043110027976443929, 7.22451300132872997519292084108, 7.79941900119667763682971898495, 8.162169145789519053179197596571, 8.534964065312342776141551775522, 8.724196582595219545465983625642, 9.420844271631511952858485504146, 9.654155887792734734389719725953

Graph of the ZZ-function along the critical line