Properties

Label 2-6003-1.1-c1-0-222
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  + 2.64·2-s + 4.97·4-s + 1.42·5-s + 2.00·7-s + 7.85·8-s + 3.75·10-s + 1.91·11-s + 5.67·13-s + 5.28·14-s + 10.8·16-s − 5.11·17-s − 0.0723·19-s + 7.07·20-s + 5.05·22-s + 23-s − 2.97·25-s + 14.9·26-s + 9.96·28-s − 29-s + 10.4·31-s + 12.8·32-s − 13.5·34-s + 2.84·35-s + 0.0359·37-s − 0.191·38-s + 11.1·40-s − 11.1·41-s + ⋯
L(s)  = 1  + 1.86·2-s + 2.48·4-s + 0.636·5-s + 0.756·7-s + 2.77·8-s + 1.18·10-s + 0.577·11-s + 1.57·13-s + 1.41·14-s + 2.70·16-s − 1.23·17-s − 0.0165·19-s + 1.58·20-s + 1.07·22-s + 0.208·23-s − 0.595·25-s + 2.94·26-s + 1.88·28-s − 0.185·29-s + 1.88·31-s + 2.26·32-s − 2.31·34-s + 0.481·35-s + 0.00590·37-s − 0.0309·38-s + 1.76·40-s − 1.74·41-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

\(L(1)\) \(\approx\) \(9.347572977\)
\(L(\frac12)\) \(\approx\) \(9.347572977\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64T + 2T^{2} \)
5 \( 1 - 1.42T + 5T^{2} \)
7 \( 1 - 2.00T + 7T^{2} \)
11 \( 1 - 1.91T + 11T^{2} \)
13 \( 1 - 5.67T + 13T^{2} \)
17 \( 1 + 5.11T + 17T^{2} \)
19 \( 1 + 0.0723T + 19T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 - 0.0359T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + 8.46T + 47T^{2} \)
53 \( 1 + 9.11T + 53T^{2} \)
59 \( 1 + 6.17T + 59T^{2} \)
61 \( 1 - 5.91T + 61T^{2} \)
67 \( 1 + 1.78T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 5.99T + 73T^{2} \)
79 \( 1 - 7.00T + 79T^{2} \)
83 \( 1 - 7.42T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + 1.32T + 97T^{2} \)
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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

−8.070464282508920755051696876369, −6.73030410293732859291031823155, −6.54099762067953043197700973443, −5.91716371582829841057010179605, −5.00026233411766494036402733055, −4.59850084056952454997007497091, −3.73236086276232399496056962304, −3.09041211593846989209494294524, −1.94947227347515778335398141662, −1.49181814521483840300029556151, 1.49181814521483840300029556151, 1.94947227347515778335398141662, 3.09041211593846989209494294524, 3.73236086276232399496056962304, 4.59850084056952454997007497091, 5.00026233411766494036402733055, 5.91716371582829841057010179605, 6.54099762067953043197700973443, 6.73030410293732859291031823155, 8.070464282508920755051696876369

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