↑

Properties

Label 4-2550e2-1.1-c1e2-0-45
Degree 44
Conductor 65025006502500
Sign 11
Analytic cond. 414.605414.605
Root an. cond. 4.512414.51241
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  + 2·2-s + 2·3-s + 3·4-s + 4·6-s + 4·8-s + 3·9-s + 6·12-s − 4·13-s + 5·16-s + 2·17-s + 6·18-s + 8·19-s + 8·23-s + 8·24-s − 8·26-s + 4·27-s + 12·29-s + 8·31-s + 6·32-s + 4·34-s + 9·36-s − 12·37-s + 16·38-s − 8·39-s + 4·41-s − 8·43-s + 16·46-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 1.41·8-s + 9-s + 1.73·12-s − 1.10·13-s + 5/4·16-s + 0.485·17-s + 1.41·18-s + 1.83·19-s + 1.66·23-s + 1.63·24-s − 1.56·26-s + 0.769·27-s + 2.22·29-s + 1.43·31-s + 1.06·32-s + 0.685·34-s + 3/2·36-s − 1.97·37-s + 2.59·38-s − 1.28·39-s + 0.624·41-s − 1.21·43-s + 2.35·46-s + ⋯

Functional equation

Λ(s)=(6502500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6502500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 65025006502500    =    22⋅32⋅54⋅1722^{2} \cdot 3^{2} \cdot 5^{4} \cdot 17^{2}
Sign: 11
Analytic conductor: 414.605414.605
Root analytic conductor: 4.512414.51241
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 6502500, ( :1/2,1/2), 1)(4,\ 6502500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 13.5753025813.57530258
L(12)L(\frac12) ≈\approx 13.5753025813.57530258
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)
bad2C1C_1 (1−T)2 ( 1 - T )^{2}
3C1C_1 (1−T)2 ( 1 - T )^{2}
5 1 1
17C1C_1 (1−T)2 ( 1 - T )^{2}
good7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13D4D_{4} 1+4T+6T2+4pT3+p2T4 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 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}
41D4D_{4} 1−4T+62T2−4pT3+p2T4 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+8T+78T2+8pT3+p2T4 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
53D4D_{4} 1+4T+14T2+4pT3+p2T4 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
61D4D_{4} 1−4T+30T2−4pT3+p2T4 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−8T+126T2−8pT3+p2T4 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+8T+134T2+8pT3+p2T4 1 + 8 T + 134 T^{2} + 8 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−12T+158T2−12pT3+p2T4 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−8T+78T2−8pT3+p2T4 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+8T+86T2+8pT3+p2T4 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−4T+86T2−4pT3+p2T4 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+4T−18T2+4pT3+p2T4 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4}
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

−9.082550147459862918676075951706, −8.606682605643484209080879198283, −8.127916647169046251029335368726, −8.062299306457441686802596270243, −7.31325713442745415679473846396, −7.18828732564724250752809734899, −6.84895881498179847375877778963, −6.50667932303110474761670697883, −5.94152664604761259414837900025, −5.21802591240586651643094799364, −5.09674292455891256716034926920, −4.90474911446282460543269684986, −4.28826888197865574242287274394, −3.81978772612130592958545511318, −3.22895280552767533646838242355, −3.08535406564464226280358519328, −2.60829502822413148291791873420, −2.28706944761092575345808084901, −1.31410150339493194386988013659, −1.01371369103375821978881681616, 1.01371369103375821978881681616, 1.31410150339493194386988013659, 2.28706944761092575345808084901, 2.60829502822413148291791873420, 3.08535406564464226280358519328, 3.22895280552767533646838242355, 3.81978772612130592958545511318, 4.28826888197865574242287274394, 4.90474911446282460543269684986, 5.09674292455891256716034926920, 5.21802591240586651643094799364, 5.94152664604761259414837900025, 6.50667932303110474761670697883, 6.84895881498179847375877778963, 7.18828732564724250752809734899, 7.31325713442745415679473846396, 8.062299306457441686802596270243, 8.127916647169046251029335368726, 8.606682605643484209080879198283, 9.082550147459862918676075951706

Graph of the ZZ-function along the critical line