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  − 1.06·2-s − 0.856·4-s + 0.420·5-s + 1.13·7-s + 3.05·8-s − 0.449·10-s − 2.91·11-s − 1.70·13-s − 1.21·14-s − 1.55·16-s + 0.536·17-s + 0.585·19-s − 0.360·20-s + 3.11·22-s + 23-s − 4.82·25-s + 1.81·26-s − 0.971·28-s + 29-s + 7.49·31-s − 4.44·32-s − 0.573·34-s + 0.477·35-s − 1.22·37-s − 0.626·38-s + 1.28·40-s + 12.2·41-s + ⋯
L(s)  = 1  − 0.756·2-s − 0.428·4-s + 0.188·5-s + 0.428·7-s + 1.07·8-s − 0.142·10-s − 0.879·11-s − 0.471·13-s − 0.324·14-s − 0.388·16-s + 0.130·17-s + 0.134·19-s − 0.0806·20-s + 0.665·22-s + 0.208·23-s − 0.964·25-s + 0.356·26-s − 0.183·28-s + 0.185·29-s + 1.34·31-s − 0.786·32-s − 0.0983·34-s + 0.0806·35-s − 0.201·37-s − 0.101·38-s + 0.203·40-s + 1.90·41-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 \( 1 + 1.06T + 2T^{2} \)
5 \( 1 - 0.420T + 5T^{2} \)
7 \( 1 - 1.13T + 7T^{2} \)
11 \( 1 + 2.91T + 11T^{2} \)
13 \( 1 + 1.70T + 13T^{2} \)
17 \( 1 - 0.536T + 17T^{2} \)
19 \( 1 - 0.585T + 19T^{2} \)
31 \( 1 - 7.49T + 31T^{2} \)
37 \( 1 + 1.22T + 37T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 + 2.78T + 43T^{2} \)
47 \( 1 - 0.180T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 + 5.86T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 4.56T + 73T^{2} \)
79 \( 1 + 6.89T + 79T^{2} \)
83 \( 1 + 8.57T + 83T^{2} \)
89 \( 1 - 2.75T + 89T^{2} \)
97 \( 1 + 7.28T + 97T^{2} \)
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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

−7.83233160858933192400508973322, −7.38755680781498496961335715671, −6.35042150471493109258493014908, −5.50464822609535095829701775238, −4.80666982074736455554649703314, −4.24666658446354029378790121115, −3.07799688548930854532430095106, −2.15226690947954985385620552200, −1.15258697518118553789476517391, 0, 1.15258697518118553789476517391, 2.15226690947954985385620552200, 3.07799688548930854532430095106, 4.24666658446354029378790121115, 4.80666982074736455554649703314, 5.50464822609535095829701775238, 6.35042150471493109258493014908, 7.38755680781498496961335715671, 7.83233160858933192400508973322

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