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  + 2.08·2-s + 2.35·4-s + 1.43·5-s − 0.274·7-s + 0.738·8-s + 2.98·10-s − 6.10·11-s + 0.188·13-s − 0.572·14-s − 3.16·16-s − 2.24·17-s + 3.65·19-s + 3.36·20-s − 12.7·22-s + 23-s − 2.95·25-s + 0.392·26-s − 0.646·28-s + 29-s − 4.05·31-s − 8.08·32-s − 4.68·34-s − 0.392·35-s − 4.16·37-s + 7.62·38-s + 1.05·40-s − 3.24·41-s + ⋯
L(s)  = 1  + 1.47·2-s + 1.17·4-s + 0.639·5-s − 0.103·7-s + 0.261·8-s + 0.943·10-s − 1.84·11-s + 0.0522·13-s − 0.153·14-s − 0.791·16-s − 0.544·17-s + 0.837·19-s + 0.752·20-s − 2.71·22-s + 0.208·23-s − 0.590·25-s + 0.0770·26-s − 0.122·28-s + 0.185·29-s − 0.728·31-s − 1.42·32-s − 0.803·34-s − 0.0663·35-s − 0.685·37-s + 1.23·38-s + 0.167·40-s − 0.506·41-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 \( 1 - 2.08T + 2T^{2} \)
5 \( 1 - 1.43T + 5T^{2} \)
7 \( 1 + 0.274T + 7T^{2} \)
11 \( 1 + 6.10T + 11T^{2} \)
13 \( 1 - 0.188T + 13T^{2} \)
17 \( 1 + 2.24T + 17T^{2} \)
19 \( 1 - 3.65T + 19T^{2} \)
31 \( 1 + 4.05T + 31T^{2} \)
37 \( 1 + 4.16T + 37T^{2} \)
41 \( 1 + 3.24T + 41T^{2} \)
43 \( 1 + 4.69T + 43T^{2} \)
47 \( 1 - 6.10T + 47T^{2} \)
53 \( 1 - 3.87T + 53T^{2} \)
59 \( 1 - 2.91T + 59T^{2} \)
61 \( 1 + 2.77T + 61T^{2} \)
67 \( 1 - 7.02T + 67T^{2} \)
71 \( 1 + 3.49T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 8.81T + 79T^{2} \)
83 \( 1 + 2.42T + 83T^{2} \)
89 \( 1 + 1.75T + 89T^{2} \)
97 \( 1 + 5.73T + 97T^{2} \)
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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

−7.46676438462376755762541623803, −6.84828606254146827014085211725, −5.97330518369653143482114603963, −5.40190452723561016195655699905, −5.04068145433847841974020054988, −4.13956261509266332487318391662, −3.21558287145799918246035968847, −2.63228002038097319642997300573, −1.80842188250684125179402193733, 0, 1.80842188250684125179402193733, 2.63228002038097319642997300573, 3.21558287145799918246035968847, 4.13956261509266332487318391662, 5.04068145433847841974020054988, 5.40190452723561016195655699905, 5.97330518369653143482114603963, 6.84828606254146827014085211725, 7.46676438462376755762541623803

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