Properties

Degree $2$
Conductor $6003$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.94·2-s + 1.78·4-s − 0.890·5-s − 3.69·7-s − 0.421·8-s − 1.73·10-s + 4.50·11-s − 2.37·13-s − 7.17·14-s − 4.38·16-s + 8.08·17-s + 3.74·19-s − 1.58·20-s + 8.75·22-s − 23-s − 4.20·25-s − 4.62·26-s − 6.58·28-s + 29-s + 0.830·31-s − 7.68·32-s + 15.7·34-s + 3.28·35-s − 6.72·37-s + 7.28·38-s + 0.375·40-s − 5.54·41-s + ⋯
L(s)  = 1  + 1.37·2-s + 0.891·4-s − 0.398·5-s − 1.39·7-s − 0.149·8-s − 0.547·10-s + 1.35·11-s − 0.659·13-s − 1.91·14-s − 1.09·16-s + 1.96·17-s + 0.858·19-s − 0.355·20-s + 1.86·22-s − 0.208·23-s − 0.841·25-s − 0.907·26-s − 1.24·28-s + 0.185·29-s + 0.149·31-s − 1.35·32-s + 2.69·34-s + 0.555·35-s − 1.10·37-s + 1.18·38-s + 0.0594·40-s − 0.866·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{6003} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6003,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 + T \)
29 \( 1 - T \)
good2 \( 1 - 1.94T + 2T^{2} \)
5 \( 1 + 0.890T + 5T^{2} \)
7 \( 1 + 3.69T + 7T^{2} \)
11 \( 1 - 4.50T + 11T^{2} \)
13 \( 1 + 2.37T + 13T^{2} \)
17 \( 1 - 8.08T + 17T^{2} \)
19 \( 1 - 3.74T + 19T^{2} \)
31 \( 1 - 0.830T + 31T^{2} \)
37 \( 1 + 6.72T + 37T^{2} \)
41 \( 1 + 5.54T + 41T^{2} \)
43 \( 1 + 4.97T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 5.72T + 53T^{2} \)
59 \( 1 - 4.08T + 59T^{2} \)
61 \( 1 + 2.66T + 61T^{2} \)
67 \( 1 + 0.373T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 0.182T + 83T^{2} \)
89 \( 1 + 0.217T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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

−7.35850076808025687571507900624, −6.86296190653110327476825164685, −6.12898457633779619965159695138, −5.57853440608039834322436315054, −4.80598158457472164409377608577, −3.83752028542601212839229293476, −3.43294643953590124908566988768, −2.91798540377670219941827839406, −1.48696621665579096522315409130, 0, 1.48696621665579096522315409130, 2.91798540377670219941827839406, 3.43294643953590124908566988768, 3.83752028542601212839229293476, 4.80598158457472164409377608577, 5.57853440608039834322436315054, 6.12898457633779619965159695138, 6.86296190653110327476825164685, 7.35850076808025687571507900624

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