Properties

Label 2-6003-1.1-c1-0-44
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.05·2-s + 2.20·4-s − 3.75·5-s − 2.35·7-s + 0.430·8-s − 7.69·10-s − 3.27·11-s + 4.29·13-s − 4.84·14-s − 3.53·16-s + 5.05·17-s − 7.69·19-s − 8.28·20-s − 6.71·22-s + 23-s + 9.07·25-s + 8.81·26-s − 5.21·28-s + 29-s − 8.24·31-s − 8.11·32-s + 10.3·34-s + 8.85·35-s − 7.63·37-s − 15.7·38-s − 1.61·40-s + 6.80·41-s + ⋯
L(s)  = 1  + 1.45·2-s + 1.10·4-s − 1.67·5-s − 0.891·7-s + 0.152·8-s − 2.43·10-s − 0.986·11-s + 1.19·13-s − 1.29·14-s − 0.884·16-s + 1.22·17-s − 1.76·19-s − 1.85·20-s − 1.43·22-s + 0.208·23-s + 1.81·25-s + 1.72·26-s − 0.985·28-s + 0.185·29-s − 1.48·31-s − 1.43·32-s + 1.77·34-s + 1.49·35-s − 1.25·37-s − 2.56·38-s − 0.255·40-s + 1.06·41-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: \(0\)
Selberg data: \((2,\ 6003,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.863660591\)
\(L(\frac12)\) \(\approx\) \(1.863660591\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.05T + 2T^{2} \)
5 \( 1 + 3.75T + 5T^{2} \)
7 \( 1 + 2.35T + 7T^{2} \)
11 \( 1 + 3.27T + 11T^{2} \)
13 \( 1 - 4.29T + 13T^{2} \)
17 \( 1 - 5.05T + 17T^{2} \)
19 \( 1 + 7.69T + 19T^{2} \)
31 \( 1 + 8.24T + 31T^{2} \)
37 \( 1 + 7.63T + 37T^{2} \)
41 \( 1 - 6.80T + 41T^{2} \)
43 \( 1 - 5.31T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 - 8.31T + 53T^{2} \)
59 \( 1 - 6.64T + 59T^{2} \)
61 \( 1 + 4.78T + 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 - 1.65T + 79T^{2} \)
83 \( 1 + 4.90T + 83T^{2} \)
89 \( 1 + 8.43T + 89T^{2} \)
97 \( 1 - 8.12T + 97T^{2} \)
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

−7.980770823062418929435409141810, −7.16195930015222248047276213141, −6.59605401279776112983189566402, −5.73244207973470314808500784094, −5.20577924142763704729792136554, −4.07616682819754024632201320990, −3.85700285550375146797464498455, −3.22800026791352467723251386521, −2.35784776078108947060045585141, −0.54537786476120804254674032466, 0.54537786476120804254674032466, 2.35784776078108947060045585141, 3.22800026791352467723251386521, 3.85700285550375146797464498455, 4.07616682819754024632201320990, 5.20577924142763704729792136554, 5.73244207973470314808500784094, 6.59605401279776112983189566402, 7.16195930015222248047276213141, 7.980770823062418929435409141810

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