Properties

Label 2-6003-1.1-c1-0-86
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.24·2-s + 3.03·4-s − 3.33·5-s + 0.336·7-s − 2.31·8-s + 7.47·10-s − 5.35·11-s − 2.46·13-s − 0.754·14-s − 0.877·16-s − 3.61·17-s − 5.53·19-s − 10.0·20-s + 12.0·22-s − 23-s + 6.10·25-s + 5.53·26-s + 1.01·28-s + 29-s + 8.47·31-s + 6.59·32-s + 8.10·34-s − 1.12·35-s + 9.55·37-s + 12.4·38-s + 7.70·40-s + 11.0·41-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.51·4-s − 1.49·5-s + 0.127·7-s − 0.817·8-s + 2.36·10-s − 1.61·11-s − 0.684·13-s − 0.201·14-s − 0.219·16-s − 0.876·17-s − 1.27·19-s − 2.25·20-s + 2.56·22-s − 0.208·23-s + 1.22·25-s + 1.08·26-s + 0.192·28-s + 0.185·29-s + 1.52·31-s + 1.16·32-s + 1.38·34-s − 0.189·35-s + 1.57·37-s + 2.01·38-s + 1.21·40-s + 1.72·41-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: $\chi_{6003} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6003,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.24T + 2T^{2} \)
5 \( 1 + 3.33T + 5T^{2} \)
7 \( 1 - 0.336T + 7T^{2} \)
11 \( 1 + 5.35T + 11T^{2} \)
13 \( 1 + 2.46T + 13T^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
19 \( 1 + 5.53T + 19T^{2} \)
31 \( 1 - 8.47T + 31T^{2} \)
37 \( 1 - 9.55T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 - 3.41T + 43T^{2} \)
47 \( 1 - 9.57T + 47T^{2} \)
53 \( 1 - 6.66T + 53T^{2} \)
59 \( 1 - 1.21T + 59T^{2} \)
61 \( 1 - 7.12T + 61T^{2} \)
67 \( 1 - 5.16T + 67T^{2} \)
71 \( 1 + 2.09T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 3.62T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 3.01T + 97T^{2} \)
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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

−7.88003633310572652608040855165, −7.41386514891966927120948919249, −6.74973234567578877527202425680, −5.77980557314294829302638463804, −4.47070354499672244849869031776, −4.32204985951967324656079172140, −2.70250782972099157010735829662, −2.37562826492444915867159441044, −0.77069239605553558716634987342, 0, 0.77069239605553558716634987342, 2.37562826492444915867159441044, 2.70250782972099157010735829662, 4.32204985951967324656079172140, 4.47070354499672244849869031776, 5.77980557314294829302638463804, 6.74973234567578877527202425680, 7.41386514891966927120948919249, 7.88003633310572652608040855165

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