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  + 1.60·2-s + 0.573·4-s − 1.49·5-s + 2.64·7-s − 2.28·8-s − 2.39·10-s + 0.452·11-s − 4.89·13-s + 4.24·14-s − 4.81·16-s + 1.33·17-s + 4.58·19-s − 0.855·20-s + 0.725·22-s + 23-s − 2.77·25-s − 7.85·26-s + 1.51·28-s + 29-s + 3.84·31-s − 3.15·32-s + 2.14·34-s − 3.94·35-s + 5.27·37-s + 7.35·38-s + 3.41·40-s − 2.68·41-s + ⋯
L(s)  = 1  + 1.13·2-s + 0.286·4-s − 0.666·5-s + 0.999·7-s − 0.808·8-s − 0.756·10-s + 0.136·11-s − 1.35·13-s + 1.13·14-s − 1.20·16-s + 0.324·17-s + 1.05·19-s − 0.191·20-s + 0.154·22-s + 0.208·23-s − 0.555·25-s − 1.54·26-s + 0.286·28-s + 0.185·29-s + 0.691·31-s − 0.557·32-s + 0.367·34-s − 0.666·35-s + 0.866·37-s + 1.19·38-s + 0.539·40-s − 0.419·41-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 \( 1 - 1.60T + 2T^{2} \)
5 \( 1 + 1.49T + 5T^{2} \)
7 \( 1 - 2.64T + 7T^{2} \)
11 \( 1 - 0.452T + 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 - 1.33T + 17T^{2} \)
19 \( 1 - 4.58T + 19T^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 - 5.27T + 37T^{2} \)
41 \( 1 + 2.68T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + 9.46T + 53T^{2} \)
59 \( 1 + 5.58T + 59T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 5.99T + 71T^{2} \)
73 \( 1 + 4.80T + 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + 2.48T + 83T^{2} \)
89 \( 1 - 1.82T + 89T^{2} \)
97 \( 1 - 5.67T + 97T^{2} \)
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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

−7.66122447003321752458455855706, −7.05850853084519670491598390944, −6.05063167212366565905316622592, −5.36907580329038440263674863369, −4.63401768415818933345145266394, −4.37993052516192171411990511137, −3.29471111444605363496264258050, −2.70586517121235767947420500871, −1.47064425424072349201002076702, 0, 1.47064425424072349201002076702, 2.70586517121235767947420500871, 3.29471111444605363496264258050, 4.37993052516192171411990511137, 4.63401768415818933345145266394, 5.36907580329038440263674863369, 6.05063167212366565905316622592, 7.05850853084519670491598390944, 7.66122447003321752458455855706

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