Properties

Degree 2
Conductor $ 3^{2} \cdot 23 \cdot 29 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.183·2-s − 1.96·4-s + 0.926·5-s − 0.385·7-s − 0.727·8-s + 0.170·10-s + 0.749·11-s + 2.20·13-s − 0.0707·14-s + 3.79·16-s − 1.34·17-s − 3.59·19-s − 1.82·20-s + 0.137·22-s + 23-s − 4.14·25-s + 0.404·26-s + 0.758·28-s + 29-s − 1.24·31-s + 2.15·32-s − 0.247·34-s − 0.357·35-s + 3.97·37-s − 0.658·38-s − 0.674·40-s − 9.73·41-s + ⋯
L(s)  = 1  + 0.129·2-s − 0.983·4-s + 0.414·5-s − 0.145·7-s − 0.257·8-s + 0.0537·10-s + 0.226·11-s + 0.610·13-s − 0.0189·14-s + 0.949·16-s − 0.326·17-s − 0.823·19-s − 0.407·20-s + 0.0293·22-s + 0.208·23-s − 0.828·25-s + 0.0792·26-s + 0.143·28-s + 0.185·29-s − 0.223·31-s + 0.380·32-s − 0.0423·34-s − 0.0604·35-s + 0.652·37-s − 0.106·38-s − 0.106·40-s − 1.51·41-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

\( d \)  =  \(2\)
\( N \)  =  \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6003} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6003,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
23 \( 1 - T \)
29 \( 1 - T \)
good2 \( 1 - 0.183T + 2T^{2} \)
5 \( 1 - 0.926T + 5T^{2} \)
7 \( 1 + 0.385T + 7T^{2} \)
11 \( 1 - 0.749T + 11T^{2} \)
13 \( 1 - 2.20T + 13T^{2} \)
17 \( 1 + 1.34T + 17T^{2} \)
19 \( 1 + 3.59T + 19T^{2} \)
31 \( 1 + 1.24T + 31T^{2} \)
37 \( 1 - 3.97T + 37T^{2} \)
41 \( 1 + 9.73T + 41T^{2} \)
43 \( 1 - 4.45T + 43T^{2} \)
47 \( 1 + 0.400T + 47T^{2} \)
53 \( 1 + 1.84T + 53T^{2} \)
59 \( 1 + 0.0802T + 59T^{2} \)
61 \( 1 - 3.33T + 61T^{2} \)
67 \( 1 - 9.55T + 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 - 7.66T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 + 2.45T + 89T^{2} \)
97 \( 1 + 0.126T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.965999608863368504716914150195, −6.86385265414216787424016785735, −6.23883579862198311228462999991, −5.55800321115119287963600515801, −4.81258350276708400368945457682, −4.04317187720064267653700045455, −3.43786223486363119221746125838, −2.31952484875978936120011763600, −1.26149258139602070093908203531, 0, 1.26149258139602070093908203531, 2.31952484875978936120011763600, 3.43786223486363119221746125838, 4.04317187720064267653700045455, 4.81258350276708400368945457682, 5.55800321115119287963600515801, 6.23883579862198311228462999991, 6.86385265414216787424016785735, 7.965999608863368504716914150195

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