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Properties

Label 4-442368-1.1-c1e2-0-14
Degree 44
Conductor 442368442368
Sign 11
Analytic cond. 28.205728.2057
Root an. cond. 2.304542.30454
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 8·11-s + 6·25-s − 27-s − 8·33-s − 2·49-s + 8·59-s + 12·73-s − 6·75-s + 81-s + 24·83-s − 28·97-s + 8·99-s + 8·107-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 2·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 2.41·11-s + 6/5·25-s − 0.192·27-s − 1.39·33-s − 2/7·49-s + 1.04·59-s + 1.40·73-s − 0.692·75-s + 1/9·81-s + 2.63·83-s − 2.84·97-s + 0.804·99-s + 0.773·107-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.164·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + ⋯

Functional equation

Λ(s)=(442368s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(442368s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 442368442368    =    214⋅332^{14} \cdot 3^{3}
Sign: 11
Analytic conductor: 28.205728.2057
Root analytic conductor: 2.304542.30454
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 442368, ( :1/2,1/2), 1)(4,\ 442368,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9149819301.914981930
L(12)L(\frac12) ≈\approx 1.9149819301.914981930
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 1 + T
good5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C22C_2^2 1−54T2+p2T4 1 - 54 T^{2} + p^{2} T^{4}
31C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{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)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
67C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
73C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
79C22C_2^2 1−142T2+p2T4 1 - 142 T^{2} + p^{2} T^{4}
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
89C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C2C_2 (1+14T+pT2)2 ( 1 + 14 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

−8.684043979652876988066025176599, −8.198948572611240427865165792362, −7.61532598445505021231566482492, −6.98089040034932339267148531426, −6.71372099189548872370259188607, −6.39335751084534575278078850353, −5.91040934760801511512508817965, −5.21950175494313500320865809431, −4.83814835762951883923045406987, −4.13901961167284522711150074983, −3.82169131902864432935982281522, −3.23290331977889648543182041186, −2.32652027208272497884774882902, −1.48465297543568880446390570490, −0.870220222855083091023826253514, 0.870220222855083091023826253514, 1.48465297543568880446390570490, 2.32652027208272497884774882902, 3.23290331977889648543182041186, 3.82169131902864432935982281522, 4.13901961167284522711150074983, 4.83814835762951883923045406987, 5.21950175494313500320865809431, 5.91040934760801511512508817965, 6.39335751084534575278078850353, 6.71372099189548872370259188607, 6.98089040034932339267148531426, 7.61532598445505021231566482492, 8.198948572611240427865165792362, 8.684043979652876988066025176599

Graph of the ZZ-function along the critical line