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  − 1.24·2-s − 0.444·4-s − 1.99·5-s − 4.26·7-s + 3.04·8-s + 2.48·10-s − 2.17·11-s + 0.563·13-s + 5.31·14-s − 2.91·16-s − 2.08·17-s + 7.02·19-s + 0.885·20-s + 2.71·22-s + 23-s − 1.02·25-s − 0.702·26-s + 1.89·28-s + 29-s − 5.22·31-s − 2.46·32-s + 2.59·34-s + 8.49·35-s − 3.21·37-s − 8.76·38-s − 6.07·40-s + 4.83·41-s + ⋯
L(s)  = 1  − 0.882·2-s − 0.222·4-s − 0.891·5-s − 1.61·7-s + 1.07·8-s + 0.786·10-s − 0.657·11-s + 0.156·13-s + 1.42·14-s − 0.728·16-s − 0.504·17-s + 1.61·19-s + 0.197·20-s + 0.579·22-s + 0.208·23-s − 0.205·25-s − 0.137·26-s + 0.357·28-s + 0.185·29-s − 0.939·31-s − 0.435·32-s + 0.445·34-s + 1.43·35-s − 0.528·37-s − 1.42·38-s − 0.960·40-s + 0.755·41-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 \( 1 + 1.24T + 2T^{2} \)
5 \( 1 + 1.99T + 5T^{2} \)
7 \( 1 + 4.26T + 7T^{2} \)
11 \( 1 + 2.17T + 11T^{2} \)
13 \( 1 - 0.563T + 13T^{2} \)
17 \( 1 + 2.08T + 17T^{2} \)
19 \( 1 - 7.02T + 19T^{2} \)
31 \( 1 + 5.22T + 31T^{2} \)
37 \( 1 + 3.21T + 37T^{2} \)
41 \( 1 - 4.83T + 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 + 6.11T + 47T^{2} \)
53 \( 1 - 8.92T + 53T^{2} \)
59 \( 1 - 6.86T + 59T^{2} \)
61 \( 1 - 1.96T + 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 - 9.69T + 71T^{2} \)
73 \( 1 - 7.14T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 6.81T + 83T^{2} \)
89 \( 1 - 5.39T + 89T^{2} \)
97 \( 1 + 3.65T + 97T^{2} \)
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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

−7.87358176509574136255151293613, −7.11096569534456421642685986551, −6.67312903707612855126834659731, −5.51879354073164817650242884696, −4.89860717000205435516633871865, −3.68609618189554026165491277125, −3.48014970962981559829164955586, −2.26648896146716600880172754810, −0.821068179556457909540779569515, 0, 0.821068179556457909540779569515, 2.26648896146716600880172754810, 3.48014970962981559829164955586, 3.68609618189554026165491277125, 4.89860717000205435516633871865, 5.51879354073164817650242884696, 6.67312903707612855126834659731, 7.11096569534456421642685986551, 7.87358176509574136255151293613

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