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Properties

Label 4-3888e2-1.1-c0e2-0-1
Degree 44
Conductor 1511654415116544
Sign 11
Analytic cond. 3.765013.76501
Root an. cond. 1.392961.39296
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  − 7-s − 2·13-s − 4·19-s − 25-s − 31-s − 2·37-s − 43-s + 49-s + 61-s − 67-s − 2·73-s + 2·79-s + 2·91-s − 2·97-s − 103-s − 2·109-s − 121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 7-s − 2·13-s − 4·19-s − 25-s − 31-s − 2·37-s − 43-s + 49-s + 61-s − 67-s − 2·73-s + 2·79-s + 2·91-s − 2·97-s − 103-s − 2·109-s − 121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯

Functional equation

Λ(s)=(15116544s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(15116544s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1511654415116544    =    28⋅3102^{8} \cdot 3^{10}
Sign: 11
Analytic conductor: 3.765013.76501
Root analytic conductor: 1.392961.39296
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 15116544, ( :0,0), 1)(4,\ 15116544,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.018148288360.01814828836
L(12)L(\frac12) ≈\approx 0.018148288360.01814828836
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)
bad2 1 1
3 1 1
good5C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
7C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
11C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
13C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
17C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
19C1C_1 (1+T)4 ( 1 + T )^{4}
23C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
29C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
31C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
37C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
41C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
43C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
47C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
53C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
59C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
61C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
67C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
71C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
73C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
79C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
83C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
89C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
97C2C_2 (1+T+T2)2 ( 1 + T + 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

−8.886808015529885984808044328872, −8.402467183063853393728792728529, −8.252037828279487126816364464705, −7.74440841320050397953276222343, −7.22939508447414411011579154741, −6.88121592284180718574654588969, −6.80933246713742258843350578098, −6.26321941633441879256100865509, −6.00489340793297580582579977728, −5.45078450852989825533104866316, −5.12933924509255600012966557161, −4.62913795408109624020504261532, −4.23420722478004079727463646490, −3.88429480254012451373613415065, −3.54376519921051401387797530369, −2.74842833700036690538867205464, −2.53149522541700497962983290257, −1.94309415356894074357782550771, −1.73211871609483328761125436127, −0.06820835859080096726495461965, 0.06820835859080096726495461965, 1.73211871609483328761125436127, 1.94309415356894074357782550771, 2.53149522541700497962983290257, 2.74842833700036690538867205464, 3.54376519921051401387797530369, 3.88429480254012451373613415065, 4.23420722478004079727463646490, 4.62913795408109624020504261532, 5.12933924509255600012966557161, 5.45078450852989825533104866316, 6.00489340793297580582579977728, 6.26321941633441879256100865509, 6.80933246713742258843350578098, 6.88121592284180718574654588969, 7.22939508447414411011579154741, 7.74440841320050397953276222343, 8.252037828279487126816364464705, 8.402467183063853393728792728529, 8.886808015529885984808044328872

Graph of the ZZ-function along the critical line