↑

Properties

Label 4-160e2-1.1-c1e2-0-5
Degree 44
Conductor 2560025600
Sign 11
Analytic cond. 1.632271.63227
Root an. cond. 1.130311.13031
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·7-s − 6·9-s + 4·17-s − 8·23-s + 25-s + 16·31-s − 12·41-s − 8·47-s + 34·49-s − 48·63-s − 12·73-s + 27·81-s − 12·89-s − 28·97-s − 8·103-s + 36·113-s + 32·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s − 64·161-s + ⋯
L(s)  = 1  + 3.02·7-s − 2·9-s + 0.970·17-s − 1.66·23-s + 1/5·25-s + 2.87·31-s − 1.87·41-s − 1.16·47-s + 34/7·49-s − 6.04·63-s − 1.40·73-s + 3·81-s − 1.27·89-s − 2.84·97-s − 0.788·103-s + 3.38·113-s + 2.93·119-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s − 5.04·161-s + ⋯

Functional equation

Λ(s)=(25600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(25600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2560025600    =    210⋅522^{10} \cdot 5^{2}
Sign: 11
Analytic conductor: 1.632271.63227
Root analytic conductor: 1.130311.13031
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 25600, ( :1/2,1/2), 1)(4,\ 25600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4410767991.441076799
L(12)L(\frac12) ≈\approx 1.4410767991.441076799
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×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{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)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{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)2 ( 1 + 4 T + p T^{2} )^{2}
29C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
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)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)2 ( 1 + 4 T + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{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)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{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}
show more
show less
   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

−10.98792460132555645103756947006, −10.12246838996141367241613997715, −9.759879680318004905765589416029, −8.521590142691914902934262227672, −8.438761280447847026288494531448, −8.187189248591461729679142296814, −7.75327442992794184642667314086, −6.81982863783073968912914119025, −5.84906736542976130410760055690, −5.61529786752672156928392151688, −4.76905661119552301505559653794, −4.55607707806255331316875895933, −3.31415635628336508949565358363, −2.39961978181641342090490014624, −1.46465509347525672726427035614, 1.46465509347525672726427035614, 2.39961978181641342090490014624, 3.31415635628336508949565358363, 4.55607707806255331316875895933, 4.76905661119552301505559653794, 5.61529786752672156928392151688, 5.84906736542976130410760055690, 6.81982863783073968912914119025, 7.75327442992794184642667314086, 8.187189248591461729679142296814, 8.438761280447847026288494531448, 8.521590142691914902934262227672, 9.759879680318004905765589416029, 10.12246838996141367241613997715, 10.98792460132555645103756947006

Graph of the ZZ-function along the critical line