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Properties

Label 4-3200-1.1-c1e2-0-2
Degree 44
Conductor 32003200
Sign 11
Analytic cond. 0.2040340.204034
Root an. cond. 0.6720870.672087
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 6·9-s − 4·13-s + 4·17-s + 3·25-s − 4·29-s + 12·37-s − 12·41-s − 12·45-s + 2·49-s + 12·53-s − 4·61-s − 8·65-s − 12·73-s + 27·81-s + 8·85-s − 12·89-s − 28·97-s + 12·101-s + 28·109-s + 36·113-s + 24·117-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s − 1.10·13-s + 0.970·17-s + 3/5·25-s − 0.742·29-s + 1.97·37-s − 1.87·41-s − 1.78·45-s + 2/7·49-s + 1.64·53-s − 0.512·61-s − 0.992·65-s − 1.40·73-s + 3·81-s + 0.867·85-s − 1.27·89-s − 2.84·97-s + 1.19·101-s + 2.68·109-s + 3.38·113-s + 2.21·117-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

Λ(s)=(3200s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3200s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 32003200    =    27⋅522^{7} \cdot 5^{2}
Sign: 11
Analytic conductor: 0.2040340.204034
Root analytic conductor: 0.6720870.672087
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 3200, ( :1/2,1/2), 1)(4,\ 3200,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.74922224540.7492222454
L(12)L(\frac12) ≈\approx 0.74922224540.7492222454
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
5C1C_1 (1−T)2 ( 1 - T )^{2}
good3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{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)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{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

−12.70951559228550877361223816630, −12.13050487476073915971902909987, −11.52426214006865205005559601963, −11.11321493608157697547696196545, −10.12246838996141367241613997715, −9.851840183892789826086741161768, −9.022035638415746178025932660186, −8.521590142691914902934262227672, −7.75327442992794184642667314086, −6.97238544391138654785050907556, −5.84906736542976130410760055690, −5.70093888866808737514223874840, −4.76905661119552301505559653794, −3.26180587408034592610487247010, −2.39961978181641342090490014624, 2.39961978181641342090490014624, 3.26180587408034592610487247010, 4.76905661119552301505559653794, 5.70093888866808737514223874840, 5.84906736542976130410760055690, 6.97238544391138654785050907556, 7.75327442992794184642667314086, 8.521590142691914902934262227672, 9.022035638415746178025932660186, 9.851840183892789826086741161768, 10.12246838996141367241613997715, 11.11321493608157697547696196545, 11.52426214006865205005559601963, 12.13050487476073915971902909987, 12.70951559228550877361223816630

Graph of the ZZ-function along the critical line