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Properties

Label 4-3344e2-1.1-c0e2-0-3
Degree 44
Conductor 1118233611182336
Sign 11
Analytic cond. 2.785132.78513
Root an. cond. 1.291841.29184
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  + 2·5-s + 2·7-s − 2·9-s + 11-s + 2·19-s + 25-s + 4·35-s + 2·43-s − 4·45-s + 49-s + 2·55-s − 4·63-s + 2·77-s + 3·81-s − 4·83-s + 4·95-s − 2·99-s − 2·125-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 2·5-s + 2·7-s − 2·9-s + 11-s + 2·19-s + 25-s + 4·35-s + 2·43-s − 4·45-s + 49-s + 2·55-s − 4·63-s + 2·77-s + 3·81-s − 4·83-s + 4·95-s − 2·99-s − 2·125-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯

Functional equation

Λ(s)=(11182336s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(11182336s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1118233611182336    =    28⋅112⋅1922^{8} \cdot 11^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 2.785132.78513
Root analytic conductor: 1.291841.29184
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 11182336, ( :0,0), 1)(4,\ 11182336,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.7647279142.764727914
L(12)L(\frac12) ≈\approx 2.7647279142.764727914
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
11C2C_2 1−T+T2 1 - T + T^{2}
19C1C_1 (1−T)2 ( 1 - T )^{2}
good3C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
5C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
7C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
13C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
17C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + 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}
31C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
37C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
41C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
43C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{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+T2)2 ( 1 + T^{2} )^{2}
61C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
67C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
71C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
73C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
79C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
83C1C_1 (1+T)4 ( 1 + T )^{4}
89C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
97C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 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

−8.993804142314755976768645742234, −8.662696738312860166938958859823, −8.301059966152248359103091616081, −7.990437356527933320210267153483, −7.39862789946841542544627402278, −7.38575284975012575810355834059, −6.53981517395737932477705907068, −6.27372223994604475169291171305, −5.78975655631155232458125327450, −5.61658216405953371288438319836, −5.26314336614174830622068832088, −5.13432971017195760823331787834, −4.35642101593796459106114158483, −4.11923576692080355485624399210, −3.22183988221379170375321930166, −3.01677768875011521869734197634, −2.27417374635565346642232489292, −2.14551292080394470732647440931, −1.29148352670585493178495700304, −1.25670040207136855628877895273, 1.25670040207136855628877895273, 1.29148352670585493178495700304, 2.14551292080394470732647440931, 2.27417374635565346642232489292, 3.01677768875011521869734197634, 3.22183988221379170375321930166, 4.11923576692080355485624399210, 4.35642101593796459106114158483, 5.13432971017195760823331787834, 5.26314336614174830622068832088, 5.61658216405953371288438319836, 5.78975655631155232458125327450, 6.27372223994604475169291171305, 6.53981517395737932477705907068, 7.38575284975012575810355834059, 7.39862789946841542544627402278, 7.990437356527933320210267153483, 8.301059966152248359103091616081, 8.662696738312860166938958859823, 8.993804142314755976768645742234

Graph of the ZZ-function along the critical line