↑

Properties

Label 4-86400-1.1-c1e2-0-14
Degree 44
Conductor 8640086400
Sign 11
Analytic cond. 5.508935.50893
Root an. cond. 1.532021.53202
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  − 3-s + 9-s + 8·11-s + 12·13-s + 25-s − 27-s − 8·33-s − 4·37-s − 12·39-s − 16·47-s − 14·49-s − 24·59-s + 28·61-s − 16·71-s − 12·73-s − 75-s + 81-s + 24·83-s + 4·97-s + 8·99-s + 8·107-s − 36·109-s + 4·111-s + 12·117-s + 26·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 2.41·11-s + 3.32·13-s + 1/5·25-s − 0.192·27-s − 1.39·33-s − 0.657·37-s − 1.92·39-s − 2.33·47-s − 2·49-s − 3.12·59-s + 3.58·61-s − 1.89·71-s − 1.40·73-s − 0.115·75-s + 1/9·81-s + 2.63·83-s + 0.406·97-s + 0.804·99-s + 0.773·107-s − 3.44·109-s + 0.379·111-s + 1.10·117-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(86400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(86400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8640086400    =    27⋅33⋅522^{7} \cdot 3^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 5.508935.50893
Root analytic conductor: 1.532021.53202
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 86400, ( :1/2,1/2), 1)(4,\ 86400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6806776381.680677638
L(12)L(\frac12) ≈\approx 1.6806776381.680677638
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
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{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}
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)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C2C_2 (1+8T+pT2)2 ( 1 + 8 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+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
61C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
67C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
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 (1−2T+pT2)2 ( 1 - 2 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

−9.487642964123562756489741043280, −9.234498555197333169762430563636, −8.663099716937980035576099714984, −8.372242663067272040741337380641, −7.76560916456884806442341251241, −6.71560882928791827542770443897, −6.58285993178476791834968599347, −6.23234003008114203357336027651, −5.74168392413099570963868387782, −4.86312932086210572949074191291, −4.19803603229033482796144552370, −3.57148112685573562185271725821, −3.38984665882479408513314051846, −1.48909737355432343830877670544, −1.36054514292875137606341288502, 1.36054514292875137606341288502, 1.48909737355432343830877670544, 3.38984665882479408513314051846, 3.57148112685573562185271725821, 4.19803603229033482796144552370, 4.86312932086210572949074191291, 5.74168392413099570963868387782, 6.23234003008114203357336027651, 6.58285993178476791834968599347, 6.71560882928791827542770443897, 7.76560916456884806442341251241, 8.372242663067272040741337380641, 8.663099716937980035576099714984, 9.234498555197333169762430563636, 9.487642964123562756489741043280

Graph of the ZZ-function along the critical line