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  − 2.33·2-s + 3.43·4-s + 2.03·5-s + 4.28·7-s − 3.34·8-s − 4.74·10-s + 3.57·11-s − 5.73·13-s − 9.98·14-s + 0.920·16-s − 4.13·17-s − 8.31·19-s + 6.99·20-s − 8.33·22-s + 23-s − 0.850·25-s + 13.3·26-s + 14.7·28-s + 29-s + 4.54·31-s + 4.53·32-s + 9.64·34-s + 8.72·35-s − 9.15·37-s + 19.3·38-s − 6.80·40-s + 3.77·41-s + ⋯
L(s)  = 1  − 1.64·2-s + 1.71·4-s + 0.910·5-s + 1.61·7-s − 1.18·8-s − 1.50·10-s + 1.07·11-s − 1.59·13-s − 2.66·14-s + 0.230·16-s − 1.00·17-s − 1.90·19-s + 1.56·20-s − 1.77·22-s + 0.208·23-s − 0.170·25-s + 2.62·26-s + 2.77·28-s + 0.185·29-s + 0.816·31-s + 0.801·32-s + 1.65·34-s + 1.47·35-s − 1.50·37-s + 3.14·38-s − 1.07·40-s + 0.589·41-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 \( 1 + 2.33T + 2T^{2} \)
5 \( 1 - 2.03T + 5T^{2} \)
7 \( 1 - 4.28T + 7T^{2} \)
11 \( 1 - 3.57T + 11T^{2} \)
13 \( 1 + 5.73T + 13T^{2} \)
17 \( 1 + 4.13T + 17T^{2} \)
19 \( 1 + 8.31T + 19T^{2} \)
31 \( 1 - 4.54T + 31T^{2} \)
37 \( 1 + 9.15T + 37T^{2} \)
41 \( 1 - 3.77T + 41T^{2} \)
43 \( 1 + 4.11T + 43T^{2} \)
47 \( 1 + 6.22T + 47T^{2} \)
53 \( 1 + 1.32T + 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 - 4.42T + 61T^{2} \)
67 \( 1 + 5.73T + 67T^{2} \)
71 \( 1 + 9.12T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 - 2.03T + 79T^{2} \)
83 \( 1 + 2.51T + 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 - 0.738T + 97T^{2} \)
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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

−8.037246679917746730084699845690, −6.99453765657428567248964253245, −6.78052016179037933212793119824, −5.78585521965876548357935516866, −4.76101043737823896656926298046, −4.29946431447811560771650229622, −2.51544210282726164682826335963, −1.97177777547200688962353415067, −1.42651356827798640288312651825, 0, 1.42651356827798640288312651825, 1.97177777547200688962353415067, 2.51544210282726164682826335963, 4.29946431447811560771650229622, 4.76101043737823896656926298046, 5.78585521965876548357935516866, 6.78052016179037933212793119824, 6.99453765657428567248964253245, 8.037246679917746730084699845690

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