Properties

Label 2-6003-1.1-c1-0-9
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.89·2-s + 1.57·4-s − 2.79·5-s + 1.18·7-s + 0.806·8-s + 5.29·10-s − 6.19·11-s + 6.24·13-s − 2.24·14-s − 4.67·16-s − 7.04·17-s − 8.14·19-s − 4.40·20-s + 11.7·22-s + 23-s + 2.83·25-s − 11.8·26-s + 1.86·28-s − 29-s − 1.41·31-s + 7.21·32-s + 13.3·34-s − 3.32·35-s + 1.97·37-s + 15.3·38-s − 2.25·40-s − 1.66·41-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.786·4-s − 1.25·5-s + 0.448·7-s + 0.285·8-s + 1.67·10-s − 1.86·11-s + 1.73·13-s − 0.599·14-s − 1.16·16-s − 1.70·17-s − 1.86·19-s − 0.984·20-s + 2.49·22-s + 0.208·23-s + 0.566·25-s − 2.31·26-s + 0.353·28-s − 0.185·29-s − 0.253·31-s + 1.27·32-s + 2.28·34-s − 0.561·35-s + 0.324·37-s + 2.49·38-s − 0.356·40-s − 0.259·41-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

\(L(1)\) \(\approx\) \(0.1480257732\)
\(L(\frac12)\) \(\approx\) \(0.1480257732\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.89T + 2T^{2} \)
5 \( 1 + 2.79T + 5T^{2} \)
7 \( 1 - 1.18T + 7T^{2} \)
11 \( 1 + 6.19T + 11T^{2} \)
13 \( 1 - 6.24T + 13T^{2} \)
17 \( 1 + 7.04T + 17T^{2} \)
19 \( 1 + 8.14T + 19T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 - 1.97T + 37T^{2} \)
41 \( 1 + 1.66T + 41T^{2} \)
43 \( 1 + 9.37T + 43T^{2} \)
47 \( 1 + 1.54T + 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 - 8.99T + 59T^{2} \)
61 \( 1 - 3.67T + 61T^{2} \)
67 \( 1 + 4.61T + 67T^{2} \)
71 \( 1 - 7.09T + 71T^{2} \)
73 \( 1 + 1.79T + 73T^{2} \)
79 \( 1 + 0.494T + 79T^{2} \)
83 \( 1 - 7.66T + 83T^{2} \)
89 \( 1 + 1.70T + 89T^{2} \)
97 \( 1 + 4.20T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.288595936629777134294613239292, −7.80232044226008030745103677246, −6.86624987282502813038156760319, −6.31600909335543340818581004984, −5.04717777751267638708017533918, −4.42430388673341389771203105141, −3.68024947841093300459842061381, −2.48386193656554289834426792352, −1.64902148312435331015754005772, −0.24415521935247141333982975998, 0.24415521935247141333982975998, 1.64902148312435331015754005772, 2.48386193656554289834426792352, 3.68024947841093300459842061381, 4.42430388673341389771203105141, 5.04717777751267638708017533918, 6.31600909335543340818581004984, 6.86624987282502813038156760319, 7.80232044226008030745103677246, 8.288595936629777134294613239292

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