Properties

Label 2-6003-1.1-c1-0-37
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  + 0.534·2-s − 1.71·4-s + 2.64·5-s − 5.04·7-s − 1.98·8-s + 1.41·10-s − 2.33·11-s + 6.26·13-s − 2.69·14-s + 2.37·16-s − 6.89·17-s − 7.12·19-s − 4.53·20-s − 1.24·22-s + 23-s + 1.99·25-s + 3.34·26-s + 8.64·28-s − 29-s − 6.09·31-s + 5.23·32-s − 3.68·34-s − 13.3·35-s − 6.24·37-s − 3.80·38-s − 5.24·40-s + 8.65·41-s + ⋯
L(s)  = 1  + 0.377·2-s − 0.857·4-s + 1.18·5-s − 1.90·7-s − 0.701·8-s + 0.446·10-s − 0.705·11-s + 1.73·13-s − 0.719·14-s + 0.592·16-s − 1.67·17-s − 1.63·19-s − 1.01·20-s − 0.266·22-s + 0.208·23-s + 0.399·25-s + 0.655·26-s + 1.63·28-s − 0.185·29-s − 1.09·31-s + 0.925·32-s − 0.631·34-s − 2.25·35-s − 1.02·37-s − 0.617·38-s − 0.829·40-s + 1.35·41-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

\(L(1)\) \(\approx\) \(1.148378532\)
\(L(\frac12)\) \(\approx\) \(1.148378532\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.534T + 2T^{2} \)
5 \( 1 - 2.64T + 5T^{2} \)
7 \( 1 + 5.04T + 7T^{2} \)
11 \( 1 + 2.33T + 11T^{2} \)
13 \( 1 - 6.26T + 13T^{2} \)
17 \( 1 + 6.89T + 17T^{2} \)
19 \( 1 + 7.12T + 19T^{2} \)
31 \( 1 + 6.09T + 31T^{2} \)
37 \( 1 + 6.24T + 37T^{2} \)
41 \( 1 - 8.65T + 41T^{2} \)
43 \( 1 - 0.338T + 43T^{2} \)
47 \( 1 - 5.85T + 47T^{2} \)
53 \( 1 - 8.40T + 53T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 1.57T + 67T^{2} \)
71 \( 1 + 3.63T + 71T^{2} \)
73 \( 1 + 5.45T + 73T^{2} \)
79 \( 1 - 7.81T + 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + 9.64T + 89T^{2} \)
97 \( 1 + 7.93T + 97T^{2} \)
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

−8.533509521718400767763690647128, −6.99511548986151626154787804281, −6.51072122317220690923013475679, −5.78033692876778637330323706429, −5.57997352840510570116670802832, −4.18232014037293060923016112542, −3.84401595838683613932116759252, −2.82250962552641071180370649624, −2.07267294920255012157920843174, −0.50716795262216186835921142715, 0.50716795262216186835921142715, 2.07267294920255012157920843174, 2.82250962552641071180370649624, 3.84401595838683613932116759252, 4.18232014037293060923016112542, 5.57997352840510570116670802832, 5.78033692876778637330323706429, 6.51072122317220690923013475679, 6.99511548986151626154787804281, 8.533509521718400767763690647128

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