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Properties

Label 4-1216e2-1.1-c1e2-0-23
Degree 44
Conductor 14786561478656
Sign −1-1
Analytic cond. 94.280394.2803
Root an. cond. 3.116053.11605
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·9-s + 2·13-s + 6·17-s − 10·25-s − 6·29-s + 20·37-s − 24·41-s − 13·49-s + 18·53-s − 20·61-s + 2·73-s − 8·89-s − 12·97-s − 20·101-s − 18·109-s + 4·113-s + 6·117-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 18·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 9-s + 0.554·13-s + 1.45·17-s − 2·25-s − 1.11·29-s + 3.28·37-s − 3.74·41-s − 1.85·49-s + 2.47·53-s − 2.56·61-s + 0.234·73-s − 0.847·89-s − 1.21·97-s − 1.99·101-s − 1.72·109-s + 0.376·113-s + 0.554·117-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.45·153-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=(1478656s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1478656s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14786561478656    =    212⋅1922^{12} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 94.280394.2803
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 1478656, ( :1/2,1/2), −1)(4,\ 1478656,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
19C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2 (1−pT+pT2)(1+pT+pT2) ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )
5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
11C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
13C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
17C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
23C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
29C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
41C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
43C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
59C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
71C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
73C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
79C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
97C2C_2 (1+6T+pT2)2 ( 1 + 6 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

−7.74236222044461134007095453218, −7.47536575646542499937872009352, −6.61303503651954034437075836396, −6.60845811420238685670203067170, −5.91824238738469752095279165720, −5.46776906443032883183491997236, −5.22289133265133441977616849928, −4.39011552338323920878309071486, −4.08931083553499496867238827018, −3.63803508541238097399374767207, −3.11827862407422559284554596552, −2.43573806798025905726670590752, −1.52291178817025335850558392010, −1.37942708011457790071691730372, 0, 1.37942708011457790071691730372, 1.52291178817025335850558392010, 2.43573806798025905726670590752, 3.11827862407422559284554596552, 3.63803508541238097399374767207, 4.08931083553499496867238827018, 4.39011552338323920878309071486, 5.22289133265133441977616849928, 5.46776906443032883183491997236, 5.91824238738469752095279165720, 6.60845811420238685670203067170, 6.61303503651954034437075836396, 7.47536575646542499937872009352, 7.74236222044461134007095453218

Graph of the ZZ-function along the critical line