Properties

Label 2-4001-1.1-c1-0-121
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  − 2.65·2-s + 3.05·3-s + 5.03·4-s − 1.94·5-s − 8.09·6-s + 2.48·7-s − 8.05·8-s + 6.32·9-s + 5.16·10-s − 2.94·11-s + 15.3·12-s + 0.192·13-s − 6.57·14-s − 5.94·15-s + 11.2·16-s + 4.76·17-s − 16.7·18-s − 2.90·19-s − 9.80·20-s + 7.57·21-s + 7.81·22-s − 3.08·23-s − 24.5·24-s − 1.20·25-s − 0.511·26-s + 10.1·27-s + 12.4·28-s + ⋯
L(s)  = 1  − 1.87·2-s + 1.76·3-s + 2.51·4-s − 0.870·5-s − 3.30·6-s + 0.937·7-s − 2.84·8-s + 2.10·9-s + 1.63·10-s − 0.887·11-s + 4.43·12-s + 0.0534·13-s − 1.75·14-s − 1.53·15-s + 2.82·16-s + 1.15·17-s − 3.95·18-s − 0.665·19-s − 2.19·20-s + 1.65·21-s + 1.66·22-s − 0.643·23-s − 5.01·24-s − 0.241·25-s − 0.100·26-s + 1.95·27-s + 2.36·28-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

\(L(1)\) \(\approx\) \(1.459610613\)
\(L(\frac12)\) \(\approx\) \(1.459610613\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.65T + 2T^{2} \)
3 \( 1 - 3.05T + 3T^{2} \)
5 \( 1 + 1.94T + 5T^{2} \)
7 \( 1 - 2.48T + 7T^{2} \)
11 \( 1 + 2.94T + 11T^{2} \)
13 \( 1 - 0.192T + 13T^{2} \)
17 \( 1 - 4.76T + 17T^{2} \)
19 \( 1 + 2.90T + 19T^{2} \)
23 \( 1 + 3.08T + 23T^{2} \)
29 \( 1 - 2.23T + 29T^{2} \)
31 \( 1 + 3.07T + 31T^{2} \)
37 \( 1 - 8.80T + 37T^{2} \)
41 \( 1 + 2.72T + 41T^{2} \)
43 \( 1 - 11.9T + 43T^{2} \)
47 \( 1 + 4.38T + 47T^{2} \)
53 \( 1 - 14.3T + 53T^{2} \)
59 \( 1 + 3.28T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 - 4.80T + 67T^{2} \)
71 \( 1 - 0.546T + 71T^{2} \)
73 \( 1 - 4.67T + 73T^{2} \)
79 \( 1 + 4.07T + 79T^{2} \)
83 \( 1 + 0.610T + 83T^{2} \)
89 \( 1 - 4.34T + 89T^{2} \)
97 \( 1 - 6.32T + 97T^{2} \)
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

−8.331111545726193065909381543637, −7.87896403307605449118993726475, −7.69331017862882415660302103349, −6.93952125214061038864168182168, −5.70003410323752246182072924079, −4.33272955523060842528632013836, −3.47060429657680253241484636214, −2.53237219329637262450601707908, −1.97049841441310583514996062681, −0.847785723882046277594212611011, 0.847785723882046277594212611011, 1.97049841441310583514996062681, 2.53237219329637262450601707908, 3.47060429657680253241484636214, 4.33272955523060842528632013836, 5.70003410323752246182072924079, 6.93952125214061038864168182168, 7.69331017862882415660302103349, 7.87896403307605449118993726475, 8.331111545726193065909381543637

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