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Properties

Label 4-117e2-1.1-c0e2-0-0
Degree 44
Conductor 1368913689
Sign 11
Analytic cond. 0.003409460.00340946
Root an. cond. 0.2416410.241641
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  − 2·7-s − 16-s − 2·19-s + 2·31-s + 2·37-s + 2·49-s − 2·67-s − 2·73-s + 2·97-s − 2·109-s + 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·7-s − 16-s − 2·19-s + 2·31-s + 2·37-s + 2·49-s − 2·67-s − 2·73-s + 2·97-s − 2·109-s + 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

Λ(s)=(13689s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(13689s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1368913689    =    34⋅1323^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 0.003409460.00340946
Root analytic conductor: 0.2416410.241641
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 13689, ( :0,0), 1)(4,\ 13689,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.31018625780.3101862578
L(12)L(\frac12) ≈\approx 0.31018625780.3101862578
L(1)L(1) 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)
bad3 1 1
13C2C_2 1+T2 1 + T^{2}
good2C22C_2^2 1+T4 1 + T^{4}
5C22C_2^2 1+T4 1 + T^{4}
7C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
11C22C_2^2 1+T4 1 + T^{4}
17C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
19C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
23C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
29C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
31C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
37C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
41C22C_2^2 1+T4 1 + T^{4}
43C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
47C22C_2^2 1+T4 1 + T^{4}
53C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
59C22C_2^2 1+T4 1 + T^{4}
61C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
67C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
71C22C_2^2 1+T4 1 + T^{4}
73C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
79C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
83C22C_2^2 1+T4 1 + T^{4}
89C22C_2^2 1+T4 1 + T^{4}
97C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + 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

−13.80990494819017442544207341531, −13.36001136281506947321561544962, −13.05270677275964737188948525381, −12.72164914596868163355269831842, −11.93499686341665259569750108108, −11.67036785348349858475122944719, −10.65262963256653539214358760704, −10.52450589542211795910627174439, −9.701049372459152198528131483308, −9.460599077616326652932631890719, −8.744582829751551591102352480621, −8.287424301189132676813928040431, −7.37991611206024721121675636931, −6.74227370788945589021286031122, −6.18811586118836933892295665132, −6.02436325627031227717949276801, −4.57945296969333583056588158114, −4.21585912692895745454493259751, −3.10156635655119368499674695018, −2.45453865405386386796231817165, 2.45453865405386386796231817165, 3.10156635655119368499674695018, 4.21585912692895745454493259751, 4.57945296969333583056588158114, 6.02436325627031227717949276801, 6.18811586118836933892295665132, 6.74227370788945589021286031122, 7.37991611206024721121675636931, 8.287424301189132676813928040431, 8.744582829751551591102352480621, 9.460599077616326652932631890719, 9.701049372459152198528131483308, 10.52450589542211795910627174439, 10.65262963256653539214358760704, 11.67036785348349858475122944719, 11.93499686341665259569750108108, 12.72164914596868163355269831842, 13.05270677275964737188948525381, 13.36001136281506947321561544962, 13.80990494819017442544207341531

Graph of the ZZ-function along the critical line