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Properties

Label 4-442368-1.1-c1e2-0-19
Degree 44
Conductor 442368442368
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·5-s + 9-s + 4·15-s − 8·19-s + 6·25-s − 27-s + 12·29-s − 8·43-s − 4·45-s + 16·47-s − 2·49-s − 4·53-s + 8·57-s + 8·67-s − 4·73-s − 6·75-s + 81-s − 12·87-s + 32·95-s + 4·97-s + 12·101-s + 10·121-s − 4·125-s + 127-s + 8·129-s + 131-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1/3·9-s + 1.03·15-s − 1.83·19-s + 6/5·25-s − 0.192·27-s + 2.22·29-s − 1.21·43-s − 0.596·45-s + 2.33·47-s − 2/7·49-s − 0.549·53-s + 1.05·57-s + 0.977·67-s − 0.468·73-s − 0.692·75-s + 1/9·81-s − 1.28·87-s + 3.28·95-s + 0.406·97-s + 1.19·101-s + 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-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: −1-1
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: 11
Selberg data: (4, 442368, ( :1/2,1/2), −1)(4,\ 442368,\ (\ :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 1 + T
good5C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
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)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×\timesC2C_2 (1−8T+pT2)(1−4T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} )
31C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−4T+pT2)(1+12T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C2C_2×\timesC2C_2 (1−12T+pT2)(1−4T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )
53C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−12T+pT2)(1+4T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 4 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×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C22C_2^2 1+114T2+p2T4 1 + 114 T^{2} + p^{2} T^{4}
83C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
89C22C_2^2 1−114T2+p2T4 1 - 114 T^{2} + p^{2} T^{4}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 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

−8.334580286466769938869095137697, −7.927228858346324339237951328163, −7.47184539419697069244714981888, −7.01102700383348905036395737363, −6.46956845524518855788592957843, −6.24990012986292931848349844623, −5.50821028045941260632498910659, −4.79025209594313437944303375568, −4.52353109771654662086830780511, −4.02631035226127556414869722600, −3.58103221487799765618724063676, −2.83760885317223549422167444783, −2.10122930606302984424679070602, −0.933135093514415288573050382372, 0, 0.933135093514415288573050382372, 2.10122930606302984424679070602, 2.83760885317223549422167444783, 3.58103221487799765618724063676, 4.02631035226127556414869722600, 4.52353109771654662086830780511, 4.79025209594313437944303375568, 5.50821028045941260632498910659, 6.24990012986292931848349844623, 6.46956845524518855788592957843, 7.01102700383348905036395737363, 7.47184539419697069244714981888, 7.927228858346324339237951328163, 8.334580286466769938869095137697

Graph of the ZZ-function along the critical line