↑

Properties

Label 2-4000-1.1-c1-0-14
Degree 22
Conductor 40004000
Sign 11
Analytic cond. 31.940131.9401
Root an. cond. 5.651565.65156
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.79·3-s + 2.30·7-s + 4.82·9-s − 5.01·11-s − 1.36·13-s + 7.56·17-s + 4.96·19-s − 6.44·21-s − 0.225·23-s − 5.09·27-s − 8.18·29-s − 6.87·31-s + 14.0·33-s + 6.20·37-s + 3.81·39-s + 4.69·41-s − 4.30·43-s + 4.39·47-s − 1.69·49-s − 21.1·51-s + 10.4·53-s − 13.8·57-s + 0.0364·59-s + 9.99·61-s + 11.1·63-s − 14.6·67-s + 0.629·69-s + ⋯
L(s)  = 1  − 1.61·3-s + 0.870·7-s + 1.60·9-s − 1.51·11-s − 0.378·13-s + 1.83·17-s + 1.13·19-s − 1.40·21-s − 0.0469·23-s − 0.981·27-s − 1.52·29-s − 1.23·31-s + 2.44·33-s + 1.02·37-s + 0.610·39-s + 0.733·41-s − 0.655·43-s + 0.641·47-s − 0.242·49-s − 2.96·51-s + 1.43·53-s − 1.83·57-s + 0.00474·59-s + 1.27·61-s + 1.39·63-s − 1.79·67-s + 0.0758·69-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 11
Analytic conductor: 31.940131.9401
Root analytic conductor: 5.651565.65156
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4000, ( :1/2), 1)(2,\ 4000,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.96002972210.9600297221
L(12)L(\frac12) ≈\approx 0.96002972210.9600297221
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}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
good3 1+2.79T+3T2 1 + 2.79T + 3T^{2}
7 1−2.30T+7T2 1 - 2.30T + 7T^{2}
11 1+5.01T+11T2 1 + 5.01T + 11T^{2}
13 1+1.36T+13T2 1 + 1.36T + 13T^{2}
17 1−7.56T+17T2 1 - 7.56T + 17T^{2}
19 1−4.96T+19T2 1 - 4.96T + 19T^{2}
23 1+0.225T+23T2 1 + 0.225T + 23T^{2}
29 1+8.18T+29T2 1 + 8.18T + 29T^{2}
31 1+6.87T+31T2 1 + 6.87T + 31T^{2}
37 1−6.20T+37T2 1 - 6.20T + 37T^{2}
41 1−4.69T+41T2 1 - 4.69T + 41T^{2}
43 1+4.30T+43T2 1 + 4.30T + 43T^{2}
47 1−4.39T+47T2 1 - 4.39T + 47T^{2}
53 1−10.4T+53T2 1 - 10.4T + 53T^{2}
59 1−0.0364T+59T2 1 - 0.0364T + 59T^{2}
61 1−9.99T+61T2 1 - 9.99T + 61T^{2}
67 1+14.6T+67T2 1 + 14.6T + 67T^{2}
71 1+1.24T+71T2 1 + 1.24T + 71T^{2}
73 1+3.30T+73T2 1 + 3.30T + 73T^{2}
79 1+8.12T+79T2 1 + 8.12T + 79T^{2}
83 1−13.8T+83T2 1 - 13.8T + 83T^{2}
89 1+0.854T+89T2 1 + 0.854T + 89T^{2}
97 1+5.51T+97T2 1 + 5.51T + 97T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.144472193589035773272705788671, −7.52969951757823987134186906716, −7.17913079797107725957699811272, −5.75490366571127792368216143016, −5.52832881642770417602979651425, −5.08509351425393120042852666914, −4.10024356054548065035827679129, −2.94983992876031083564825909209, −1.67617333844947731914245693546, −0.62782474698862431601926461297, 0.62782474698862431601926461297, 1.67617333844947731914245693546, 2.94983992876031083564825909209, 4.10024356054548065035827679129, 5.08509351425393120042852666914, 5.52832881642770417602979651425, 5.75490366571127792368216143016, 7.17913079797107725957699811272, 7.52969951757823987134186906716, 8.144472193589035773272705788671

Graph of the ZZ-function along the critical line