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Properties

Label 4-43904-1.1-c1e2-0-6
Degree 44
Conductor 4390443904
Sign −1-1
Analytic cond. 2.799352.79935
Root an. cond. 1.293491.29349
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 7-s + 6·9-s + 4·19-s + 4·21-s + 6·25-s + 4·27-s + 4·29-s − 8·31-s − 12·37-s + 8·47-s + 49-s − 20·53-s − 16·57-s − 12·59-s − 6·63-s − 24·75-s − 37·81-s − 12·83-s − 16·87-s + 32·93-s + 24·103-s + 20·109-s + 48·111-s + 12·113-s − 22·121-s + 127-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.377·7-s + 2·9-s + 0.917·19-s + 0.872·21-s + 6/5·25-s + 0.769·27-s + 0.742·29-s − 1.43·31-s − 1.97·37-s + 1.16·47-s + 1/7·49-s − 2.74·53-s − 2.11·57-s − 1.56·59-s − 0.755·63-s − 2.77·75-s − 4.11·81-s − 1.31·83-s − 1.71·87-s + 3.31·93-s + 2.36·103-s + 1.91·109-s + 4.55·111-s + 1.12·113-s − 2·121-s + 0.0887·127-s + ⋯

Functional equation

Λ(s)=(43904s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(43904s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4390443904    =    27⋅732^{7} \cdot 7^{3}
Sign: −1-1
Analytic conductor: 2.799352.79935
Root analytic conductor: 1.293491.29349
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 43904, ( :1/2,1/2), −1)(4,\ 43904,\ (\ :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
7C1C_1 1+T 1 + T
good3C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
31C2C_2 (1+4T+pT2)2 ( 1 + 4 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 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
53C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
59C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
61C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 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+pT2)2 ( 1 + p T^{2} )^{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+6T+pT2)2 ( 1 + 6 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 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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

−10.27322067133513101779746201704, −9.590662754018899629342276160434, −8.870187268949863787955145092048, −8.605229056688940905430426859414, −7.48523560915642098619060234168, −7.22672629765934394974316502798, −6.34948564084072494191115955004, −6.27767574855681654030919817110, −5.57466416223045488855827312533, −4.90170998996837834018405216678, −4.88658861407619853654072156509, −3.61118272831897977078492843408, −2.87703403465824901690577931359, −1.30690906969982341922250798482, 0, 1.30690906969982341922250798482, 2.87703403465824901690577931359, 3.61118272831897977078492843408, 4.88658861407619853654072156509, 4.90170998996837834018405216678, 5.57466416223045488855827312533, 6.27767574855681654030919817110, 6.34948564084072494191115955004, 7.22672629765934394974316502798, 7.48523560915642098619060234168, 8.605229056688940905430426859414, 8.870187268949863787955145092048, 9.590662754018899629342276160434, 10.27322067133513101779746201704

Graph of the ZZ-function along the critical line