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Properties

Label 4-3724e2-1.1-c0e2-0-2
Degree 44
Conductor 1386817613868176
Sign 11
Analytic cond. 3.454083.45408
Root an. cond. 1.363271.36327
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  + 5-s − 9-s + 11-s + 17-s − 19-s − 2·23-s + 25-s − 2·43-s − 45-s + 47-s + 55-s + 61-s + 73-s + 4·83-s + 85-s − 95-s − 99-s − 2·101-s − 2·115-s + 121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 5-s − 9-s + 11-s + 17-s − 19-s − 2·23-s + 25-s − 2·43-s − 45-s + 47-s + 55-s + 61-s + 73-s + 4·83-s + 85-s − 95-s − 99-s − 2·101-s − 2·115-s + 121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯

Functional equation

Λ(s)=(13868176s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(13868176s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1386817613868176    =    24⋅74⋅1922^{4} \cdot 7^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 3.454083.45408
Root analytic conductor: 1.363271.36327
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 13868176, ( :0,0), 1)(4,\ 13868176,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5958070691.595807069
L(12)L(\frac12) ≈\approx 1.5958070691.595807069
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
7 1 1
19C2C_2 1+T+T2 1 + T + T^{2}
good3C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
5C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
11C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
13C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
17C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
23C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
29C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
31C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
37C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
41C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
43C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
47C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
53C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + 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} )
67C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
71C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
73C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
79C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
83C1C_1 (1−T)4 ( 1 - T )^{4}
89C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + 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

−8.717109917227330457579181591885, −8.699629564000920235757333711119, −8.124982455574306957142741798460, −7.909236072443285415643004601586, −7.50783802004575073445301186780, −6.81166801410978967905108111328, −6.48957887246313796201305956739, −6.43768228897130868959087639998, −5.85600528650101313618957763664, −5.61881413269342878491488954252, −5.23116276040291331517536910185, −4.81419628247920123479889106032, −4.23782174999867611170028495022, −3.81565440144836153143760433219, −3.47791715399893294117677873613, −2.97264276023836715606033146379, −2.34453309674846738681748061847, −2.01209835935887637317845673145, −1.56922280159680543184160473022, −0.72593903728398656606338243250, 0.72593903728398656606338243250, 1.56922280159680543184160473022, 2.01209835935887637317845673145, 2.34453309674846738681748061847, 2.97264276023836715606033146379, 3.47791715399893294117677873613, 3.81565440144836153143760433219, 4.23782174999867611170028495022, 4.81419628247920123479889106032, 5.23116276040291331517536910185, 5.61881413269342878491488954252, 5.85600528650101313618957763664, 6.43768228897130868959087639998, 6.48957887246313796201305956739, 6.81166801410978967905108111328, 7.50783802004575073445301186780, 7.909236072443285415643004601586, 8.124982455574306957142741798460, 8.699629564000920235757333711119, 8.717109917227330457579181591885

Graph of the ZZ-function along the critical line