Properties

Label 2-6003-1.1-c1-0-170
Degree $2$
Conductor $6003$
Sign $-1$
Analytic cond. $47.9341$
Root an. cond. $6.92345$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.586·2-s − 1.65·4-s + 0.842·5-s + 1.23·7-s + 2.14·8-s − 0.494·10-s − 1.47·11-s − 3.26·13-s − 0.725·14-s + 2.05·16-s + 4.93·17-s + 0.260·19-s − 1.39·20-s + 0.861·22-s − 23-s − 4.28·25-s + 1.91·26-s − 2.05·28-s + 29-s − 0.101·31-s − 5.49·32-s − 2.89·34-s + 1.04·35-s − 3.33·37-s − 0.152·38-s + 1.80·40-s + 0.732·41-s + ⋯
L(s)  = 1  − 0.414·2-s − 0.828·4-s + 0.376·5-s + 0.468·7-s + 0.757·8-s − 0.156·10-s − 0.443·11-s − 0.904·13-s − 0.194·14-s + 0.514·16-s + 1.19·17-s + 0.0598·19-s − 0.312·20-s + 0.183·22-s − 0.208·23-s − 0.857·25-s + 0.374·26-s − 0.387·28-s + 0.185·29-s − 0.0182·31-s − 0.970·32-s − 0.496·34-s + 0.176·35-s − 0.548·37-s − 0.0248·38-s + 0.285·40-s + 0.114·41-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: $-1$
Analytic conductor: \(47.9341\)
Root analytic conductor: \(6.92345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
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 \( 1 + 0.586T + 2T^{2} \)
5 \( 1 - 0.842T + 5T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 + 1.47T + 11T^{2} \)
13 \( 1 + 3.26T + 13T^{2} \)
17 \( 1 - 4.93T + 17T^{2} \)
19 \( 1 - 0.260T + 19T^{2} \)
31 \( 1 + 0.101T + 31T^{2} \)
37 \( 1 + 3.33T + 37T^{2} \)
41 \( 1 - 0.732T + 41T^{2} \)
43 \( 1 + 5.01T + 43T^{2} \)
47 \( 1 - 8.71T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 8.76T + 59T^{2} \)
61 \( 1 + 1.88T + 61T^{2} \)
67 \( 1 + 3.21T + 67T^{2} \)
71 \( 1 - 1.72T + 71T^{2} \)
73 \( 1 + 2.64T + 73T^{2} \)
79 \( 1 - 0.465T + 79T^{2} \)
83 \( 1 - 8.34T + 83T^{2} \)
89 \( 1 - 5.13T + 89T^{2} \)
97 \( 1 + 6.87T + 97T^{2} \)
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

−7.73375833751198043393590579003, −7.39520098198492210452282309649, −6.22109087730296851571216590440, −5.38966346747140602776143224801, −4.98245035264754679482117611234, −4.12939245089958180620831937189, −3.24018649433003913302487903216, −2.16126485165231768044948467772, −1.22537679002315713968916008926, 0, 1.22537679002315713968916008926, 2.16126485165231768044948467772, 3.24018649433003913302487903216, 4.12939245089958180620831937189, 4.98245035264754679482117611234, 5.38966346747140602776143224801, 6.22109087730296851571216590440, 7.39520098198492210452282309649, 7.73375833751198043393590579003

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