Properties

Label 2-6003-1.1-c1-0-154
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  − 0.161·2-s − 1.97·4-s − 2.08·5-s + 3.66·7-s + 0.641·8-s + 0.335·10-s − 3.77·11-s + 3.51·13-s − 0.592·14-s + 3.84·16-s − 3.02·17-s − 1.97·19-s + 4.10·20-s + 0.609·22-s + 23-s − 0.669·25-s − 0.566·26-s − 7.24·28-s − 29-s − 3.58·31-s − 1.90·32-s + 0.488·34-s − 7.63·35-s + 5.34·37-s + 0.318·38-s − 1.33·40-s + 12.0·41-s + ⋯
L(s)  = 1  − 0.114·2-s − 0.986·4-s − 0.930·5-s + 1.38·7-s + 0.226·8-s + 0.106·10-s − 1.13·11-s + 0.973·13-s − 0.158·14-s + 0.961·16-s − 0.733·17-s − 0.453·19-s + 0.918·20-s + 0.130·22-s + 0.208·23-s − 0.133·25-s − 0.111·26-s − 1.36·28-s − 0.185·29-s − 0.643·31-s − 0.336·32-s + 0.0837·34-s − 1.29·35-s + 0.878·37-s + 0.0517·38-s − 0.211·40-s + 1.87·41-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 \( 1 + 0.161T + 2T^{2} \)
5 \( 1 + 2.08T + 5T^{2} \)
7 \( 1 - 3.66T + 7T^{2} \)
11 \( 1 + 3.77T + 11T^{2} \)
13 \( 1 - 3.51T + 13T^{2} \)
17 \( 1 + 3.02T + 17T^{2} \)
19 \( 1 + 1.97T + 19T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 - 5.34T + 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 - 3.68T + 43T^{2} \)
47 \( 1 - 0.138T + 47T^{2} \)
53 \( 1 + 3.01T + 53T^{2} \)
59 \( 1 + 9.43T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 4.57T + 67T^{2} \)
71 \( 1 - 3.28T + 71T^{2} \)
73 \( 1 - 1.53T + 73T^{2} \)
79 \( 1 - 7.31T + 79T^{2} \)
83 \( 1 + 5.82T + 83T^{2} \)
89 \( 1 + 6.06T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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

−7.75326783050617329829955420241, −7.52078791863272783245059518642, −6.14768637801084131679208013051, −5.45530089619803093070510781278, −4.56985746077452200771931595298, −4.32301350655711589936875213048, −3.42386088399910151596128410449, −2.26301968237805607199296893701, −1.13017656210983374175216924077, 0, 1.13017656210983374175216924077, 2.26301968237805607199296893701, 3.42386088399910151596128410449, 4.32301350655711589936875213048, 4.56985746077452200771931595298, 5.45530089619803093070510781278, 6.14768637801084131679208013051, 7.52078791863272783245059518642, 7.75326783050617329829955420241

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