Properties

Label 2-4001-1.1-c1-0-132
Degree $2$
Conductor $4001$
Sign $1$
Analytic cond. $31.9481$
Root an. cond. $5.65226$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.635·2-s + 2.40·3-s − 1.59·4-s − 0.176·5-s + 1.52·6-s + 0.607·7-s − 2.28·8-s + 2.79·9-s − 0.111·10-s + 0.724·11-s − 3.84·12-s − 3.12·13-s + 0.385·14-s − 0.424·15-s + 1.74·16-s − 1.81·17-s + 1.77·18-s + 3.76·19-s + 0.281·20-s + 1.46·21-s + 0.459·22-s + 4.76·23-s − 5.49·24-s − 4.96·25-s − 1.98·26-s − 0.488·27-s − 0.970·28-s + ⋯
L(s)  = 1  + 0.449·2-s + 1.39·3-s − 0.798·4-s − 0.0787·5-s + 0.624·6-s + 0.229·7-s − 0.807·8-s + 0.932·9-s − 0.0353·10-s + 0.218·11-s − 1.10·12-s − 0.866·13-s + 0.103·14-s − 0.109·15-s + 0.435·16-s − 0.440·17-s + 0.418·18-s + 0.864·19-s + 0.0628·20-s + 0.319·21-s + 0.0980·22-s + 0.993·23-s − 1.12·24-s − 0.993·25-s − 0.389·26-s − 0.0939·27-s − 0.183·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4001\)
Sign: $1$
Analytic conductor: \(31.9481\)
Root analytic conductor: \(5.65226\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.166113003\)
\(L(\frac12)\) \(\approx\) \(3.166113003\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1+O(T) \)
good2 \( 1 - 0.635T + 2T^{2} \)
3 \( 1 - 2.40T + 3T^{2} \)
5 \( 1 + 0.176T + 5T^{2} \)
7 \( 1 - 0.607T + 7T^{2} \)
11 \( 1 - 0.724T + 11T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 + 1.81T + 17T^{2} \)
19 \( 1 - 3.76T + 19T^{2} \)
23 \( 1 - 4.76T + 23T^{2} \)
29 \( 1 - 9.17T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + 4.97T + 37T^{2} \)
41 \( 1 - 6.40T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 + 8.43T + 47T^{2} \)
53 \( 1 + 1.37T + 53T^{2} \)
59 \( 1 + 3.41T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 - 3.00T + 67T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 - 7.69T + 73T^{2} \)
79 \( 1 + 6.27T + 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + 0.176T + 89T^{2} \)
97 \( 1 - 8.88T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.221427166135394648595867863359, −8.127416957150230157375570345864, −7.11504994753222242353083083768, −6.25248023309007550947657730659, −5.14469145856837819154476398624, −4.59870458424750377646260137709, −3.79947617294834671662097690393, −2.99496538104694308719629905951, −2.37442417410358334100929657818, −0.913941589373625120554315972341, 0.913941589373625120554315972341, 2.37442417410358334100929657818, 2.99496538104694308719629905951, 3.79947617294834671662097690393, 4.59870458424750377646260137709, 5.14469145856837819154476398624, 6.25248023309007550947657730659, 7.11504994753222242353083083768, 8.127416957150230157375570345864, 8.221427166135394648595867863359

Graph of the $Z$-function along the critical line