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  + 2.60·2-s + 4.77·4-s − 2.74·5-s − 2.14·7-s + 7.23·8-s − 7.15·10-s + 0.935·11-s − 3.80·13-s − 5.58·14-s + 9.26·16-s + 0.395·17-s + 3.48·19-s − 13.1·20-s + 2.43·22-s + 23-s + 2.55·25-s − 9.91·26-s − 10.2·28-s + 29-s − 6.61·31-s + 9.67·32-s + 1.02·34-s + 5.89·35-s − 9.78·37-s + 9.08·38-s − 19.8·40-s − 9.64·41-s + ⋯
L(s)  = 1  + 1.84·2-s + 2.38·4-s − 1.22·5-s − 0.810·7-s + 2.55·8-s − 2.26·10-s + 0.281·11-s − 1.05·13-s − 1.49·14-s + 2.31·16-s + 0.0958·17-s + 0.800·19-s − 2.93·20-s + 0.519·22-s + 0.208·23-s + 0.511·25-s − 1.94·26-s − 1.93·28-s + 0.185·29-s − 1.18·31-s + 1.70·32-s + 0.176·34-s + 0.996·35-s − 1.60·37-s + 1.47·38-s − 3.14·40-s − 1.50·41-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 \( 1 - 2.60T + 2T^{2} \)
5 \( 1 + 2.74T + 5T^{2} \)
7 \( 1 + 2.14T + 7T^{2} \)
11 \( 1 - 0.935T + 11T^{2} \)
13 \( 1 + 3.80T + 13T^{2} \)
17 \( 1 - 0.395T + 17T^{2} \)
19 \( 1 - 3.48T + 19T^{2} \)
31 \( 1 + 6.61T + 31T^{2} \)
37 \( 1 + 9.78T + 37T^{2} \)
41 \( 1 + 9.64T + 41T^{2} \)
43 \( 1 - 7.07T + 43T^{2} \)
47 \( 1 + 6.39T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 - 9.19T + 61T^{2} \)
67 \( 1 + 0.642T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 1.05T + 73T^{2} \)
79 \( 1 - 3.20T + 79T^{2} \)
83 \( 1 + 0.860T + 83T^{2} \)
89 \( 1 - 2.79T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−7.40846825869724186405171769984, −6.87413633528585424027403969628, −6.28159679890738204778996446260, −5.23099090078532283409903905980, −4.90274235618057916173969968994, −3.91836773692749726605054037600, −3.44753660285780656176266471982, −2.88872727800454842980602848370, −1.73740181225608367248466492218, 0, 1.73740181225608367248466492218, 2.88872727800454842980602848370, 3.44753660285780656176266471982, 3.91836773692749726605054037600, 4.90274235618057916173969968994, 5.23099090078532283409903905980, 6.28159679890738204778996446260, 6.87413633528585424027403969628, 7.40846825869724186405171769984

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