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Properties

Label 4-395136-1.1-c1e2-0-42
Degree 44
Conductor 395136395136
Sign −1-1
Analytic cond. 25.194225.1942
Root an. cond. 2.240392.24039
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 7-s + 3·9-s − 8·19-s − 2·21-s − 6·25-s + 4·27-s − 20·29-s + 16·31-s + 12·37-s − 16·47-s + 49-s − 20·53-s − 16·57-s − 24·59-s − 3·63-s − 12·75-s + 5·81-s − 24·83-s − 40·87-s + 32·93-s − 4·109-s + 24·111-s + 36·113-s − 22·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.377·7-s + 9-s − 1.83·19-s − 0.436·21-s − 6/5·25-s + 0.769·27-s − 3.71·29-s + 2.87·31-s + 1.97·37-s − 2.33·47-s + 1/7·49-s − 2.74·53-s − 2.11·57-s − 3.12·59-s − 0.377·63-s − 1.38·75-s + 5/9·81-s − 2.63·83-s − 4.28·87-s + 3.31·93-s − 0.383·109-s + 2.27·111-s + 3.38·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(395136s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(395136s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 395136395136    =    27⋅32⋅732^{7} \cdot 3^{2} \cdot 7^{3}
Sign: −1-1
Analytic conductor: 25.194225.1942
Root analytic conductor: 2.240392.24039
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 395136, ( :1/2,1/2), −1)(4,\ 395136,\ (\ :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
3C1C_1 (1−T)2 ( 1 - T )^{2}
7C1C_1 1+T 1 + T
good5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
41C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
53C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
59C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
89C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
97C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p 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

−8.289231676437400116460418770344, −7.917190719920927542416951015280, −7.84194967359953188564623556072, −7.13572102783324715252694770569, −6.54428519689900750892915035611, −6.05800461819552664757143425288, −5.90015243373728427076887013140, −4.81364713899235799057960760690, −4.35622436888448856514855173618, −4.09990139504038953260193041491, −3.10934300001651305626809863177, −3.06416831077517725944120442154, −1.98451470429642242976214434683, −1.70572377211895013061556457420, 0, 1.70572377211895013061556457420, 1.98451470429642242976214434683, 3.06416831077517725944120442154, 3.10934300001651305626809863177, 4.09990139504038953260193041491, 4.35622436888448856514855173618, 4.81364713899235799057960760690, 5.90015243373728427076887013140, 6.05800461819552664757143425288, 6.54428519689900750892915035611, 7.13572102783324715252694770569, 7.84194967359953188564623556072, 7.917190719920927542416951015280, 8.289231676437400116460418770344

Graph of the ZZ-function along the critical line