Properties

Label 2-6003-1.1-c1-0-117
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.959·2-s − 1.07·4-s + 2.76·5-s + 2.95·7-s + 2.95·8-s − 2.64·10-s − 1.28·11-s + 5.35·13-s − 2.83·14-s − 0.675·16-s + 5.11·17-s − 7.15·19-s − 2.98·20-s + 1.22·22-s + 23-s + 2.62·25-s − 5.14·26-s − 3.18·28-s − 29-s + 6.82·31-s − 5.26·32-s − 4.91·34-s + 8.15·35-s + 1.42·37-s + 6.86·38-s + 8.15·40-s + 0.739·41-s + ⋯
L(s)  = 1  − 0.678·2-s − 0.539·4-s + 1.23·5-s + 1.11·7-s + 1.04·8-s − 0.837·10-s − 0.386·11-s + 1.48·13-s − 0.757·14-s − 0.168·16-s + 1.24·17-s − 1.64·19-s − 0.666·20-s + 0.262·22-s + 0.208·23-s + 0.524·25-s − 1.00·26-s − 0.602·28-s − 0.185·29-s + 1.22·31-s − 0.930·32-s − 0.842·34-s + 1.37·35-s + 0.234·37-s + 1.11·38-s + 1.28·40-s + 0.115·41-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

\(L(1)\) \(\approx\) \(2.026825952\)
\(L(\frac12)\) \(\approx\) \(2.026825952\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.959T + 2T^{2} \)
5 \( 1 - 2.76T + 5T^{2} \)
7 \( 1 - 2.95T + 7T^{2} \)
11 \( 1 + 1.28T + 11T^{2} \)
13 \( 1 - 5.35T + 13T^{2} \)
17 \( 1 - 5.11T + 17T^{2} \)
19 \( 1 + 7.15T + 19T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 - 1.42T + 37T^{2} \)
41 \( 1 - 0.739T + 41T^{2} \)
43 \( 1 - 8.76T + 43T^{2} \)
47 \( 1 + 4.04T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 - 1.57T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 - 8.17T + 67T^{2} \)
71 \( 1 - 8.05T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 + 4.10T + 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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

−8.128143957707358313614604184657, −7.78054706754248768317355888919, −6.59391066793036995852177150657, −5.90071260591055631983346114295, −5.29503047766792720935361334081, −4.52085017972384659597064668508, −3.76310134304346775057357190443, −2.45279879858140484219492290229, −1.60177176416839697905617498584, −0.931804509630557166833825478700, 0.931804509630557166833825478700, 1.60177176416839697905617498584, 2.45279879858140484219492290229, 3.76310134304346775057357190443, 4.52085017972384659597064668508, 5.29503047766792720935361334081, 5.90071260591055631983346114295, 6.59391066793036995852177150657, 7.78054706754248768317355888919, 8.128143957707358313614604184657

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