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  − 2.24·2-s + 3.03·4-s + 2.70·5-s − 1.81·7-s − 2.33·8-s − 6.08·10-s + 1.28·11-s + 4.53·13-s + 4.07·14-s − 0.841·16-s − 4.48·17-s + 2.67·19-s + 8.23·20-s − 2.89·22-s + 23-s + 2.33·25-s − 10.1·26-s − 5.51·28-s + 29-s − 5.79·31-s + 6.55·32-s + 10.0·34-s − 4.91·35-s − 5.91·37-s − 5.99·38-s − 6.31·40-s − 7.82·41-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.51·4-s + 1.21·5-s − 0.686·7-s − 0.824·8-s − 1.92·10-s + 0.388·11-s + 1.25·13-s + 1.08·14-s − 0.210·16-s − 1.08·17-s + 0.613·19-s + 1.84·20-s − 0.616·22-s + 0.208·23-s + 0.467·25-s − 1.99·26-s − 1.04·28-s + 0.185·29-s − 1.04·31-s + 1.15·32-s + 1.72·34-s − 0.831·35-s − 0.973·37-s − 0.973·38-s − 0.999·40-s − 1.22·41-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 \( 1 + 2.24T + 2T^{2} \)
5 \( 1 - 2.70T + 5T^{2} \)
7 \( 1 + 1.81T + 7T^{2} \)
11 \( 1 - 1.28T + 11T^{2} \)
13 \( 1 - 4.53T + 13T^{2} \)
17 \( 1 + 4.48T + 17T^{2} \)
19 \( 1 - 2.67T + 19T^{2} \)
31 \( 1 + 5.79T + 31T^{2} \)
37 \( 1 + 5.91T + 37T^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 + 5.28T + 43T^{2} \)
47 \( 1 - 2.52T + 47T^{2} \)
53 \( 1 - 7.14T + 53T^{2} \)
59 \( 1 + 7.87T + 59T^{2} \)
61 \( 1 - 0.0192T + 61T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 - 8.18T + 71T^{2} \)
73 \( 1 + 16.8T + 73T^{2} \)
79 \( 1 - 2.32T + 79T^{2} \)
83 \( 1 - 8.65T + 83T^{2} \)
89 \( 1 - 17.6T + 89T^{2} \)
97 \( 1 + 7.50T + 97T^{2} \)
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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

−7.905868653418773184097557830094, −6.96872351534618296698441869540, −6.54661885085538429116882954472, −5.94676779093067338062372085369, −5.04662227909735942253323892764, −3.83170049517059467314550131931, −2.90324061930866283576997144192, −1.86516315426915866092447823202, −1.34287562312540155761585199622, 0, 1.34287562312540155761585199622, 1.86516315426915866092447823202, 2.90324061930866283576997144192, 3.83170049517059467314550131931, 5.04662227909735942253323892764, 5.94676779093067338062372085369, 6.54661885085538429116882954472, 6.96872351534618296698441869540, 7.905868653418773184097557830094

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