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  + 2.31·2-s + 3.36·4-s − 2.07·5-s + 1.30·7-s + 3.16·8-s − 4.81·10-s − 0.169·11-s + 0.705·13-s + 3.02·14-s + 0.600·16-s − 2.81·17-s − 7.73·19-s − 6.99·20-s − 0.392·22-s + 23-s − 0.686·25-s + 1.63·26-s + 4.38·28-s + 29-s − 4.11·31-s − 4.94·32-s − 6.51·34-s − 2.70·35-s + 4.93·37-s − 17.9·38-s − 6.57·40-s − 2.38·41-s + ⋯
L(s)  = 1  + 1.63·2-s + 1.68·4-s − 0.928·5-s + 0.492·7-s + 1.11·8-s − 1.52·10-s − 0.0511·11-s + 0.195·13-s + 0.807·14-s + 0.150·16-s − 0.682·17-s − 1.77·19-s − 1.56·20-s − 0.0837·22-s + 0.208·23-s − 0.137·25-s + 0.320·26-s + 0.829·28-s + 0.185·29-s − 0.738·31-s − 0.873·32-s − 1.11·34-s − 0.457·35-s + 0.811·37-s − 2.90·38-s − 1.03·40-s − 0.372·41-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 \( 1 - 2.31T + 2T^{2} \)
5 \( 1 + 2.07T + 5T^{2} \)
7 \( 1 - 1.30T + 7T^{2} \)
11 \( 1 + 0.169T + 11T^{2} \)
13 \( 1 - 0.705T + 13T^{2} \)
17 \( 1 + 2.81T + 17T^{2} \)
19 \( 1 + 7.73T + 19T^{2} \)
31 \( 1 + 4.11T + 31T^{2} \)
37 \( 1 - 4.93T + 37T^{2} \)
41 \( 1 + 2.38T + 41T^{2} \)
43 \( 1 + 1.36T + 43T^{2} \)
47 \( 1 + 8.06T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 - 5.91T + 59T^{2} \)
61 \( 1 + 2.67T + 61T^{2} \)
67 \( 1 + 4.10T + 67T^{2} \)
71 \( 1 - 2.86T + 71T^{2} \)
73 \( 1 + 9.80T + 73T^{2} \)
79 \( 1 + 3.16T + 79T^{2} \)
83 \( 1 + 4.01T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + 8.28T + 97T^{2} \)
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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

−7.49752265794993751182576070762, −6.80967557471719784160645046939, −6.20171097458393720668795769843, −5.42143846625154843714509304996, −4.56604734534359024437293883290, −4.22120187570615041196569683076, −3.52864487958875550646030392310, −2.59866969930534299465780487667, −1.76398297602110708434762341913, 0, 1.76398297602110708434762341913, 2.59866969930534299465780487667, 3.52864487958875550646030392310, 4.22120187570615041196569683076, 4.56604734534359024437293883290, 5.42143846625154843714509304996, 6.20171097458393720668795769843, 6.80967557471719784160645046939, 7.49752265794993751182576070762

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