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  − 1.35·2-s − 0.150·4-s + 4.20·5-s + 0.870·7-s + 2.92·8-s − 5.71·10-s − 3.24·11-s − 1.60·13-s − 1.18·14-s − 3.67·16-s − 7.96·17-s − 1.89·19-s − 0.632·20-s + 4.40·22-s + 23-s + 12.6·25-s + 2.17·26-s − 0.131·28-s + 29-s − 6.87·31-s − 0.850·32-s + 10.8·34-s + 3.65·35-s + 5.65·37-s + 2.57·38-s + 12.2·40-s + 5.17·41-s + ⋯
L(s)  = 1  − 0.961·2-s − 0.0753·4-s + 1.87·5-s + 0.328·7-s + 1.03·8-s − 1.80·10-s − 0.977·11-s − 0.444·13-s − 0.316·14-s − 0.918·16-s − 1.93·17-s − 0.434·19-s − 0.141·20-s + 0.940·22-s + 0.208·23-s + 2.52·25-s + 0.427·26-s − 0.0247·28-s + 0.185·29-s − 1.23·31-s − 0.150·32-s + 1.85·34-s + 0.617·35-s + 0.929·37-s + 0.417·38-s + 1.94·40-s + 0.807·41-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 \( 1 + 1.35T + 2T^{2} \)
5 \( 1 - 4.20T + 5T^{2} \)
7 \( 1 - 0.870T + 7T^{2} \)
11 \( 1 + 3.24T + 11T^{2} \)
13 \( 1 + 1.60T + 13T^{2} \)
17 \( 1 + 7.96T + 17T^{2} \)
19 \( 1 + 1.89T + 19T^{2} \)
31 \( 1 + 6.87T + 31T^{2} \)
37 \( 1 - 5.65T + 37T^{2} \)
41 \( 1 - 5.17T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 - 4.23T + 47T^{2} \)
53 \( 1 + 6.69T + 53T^{2} \)
59 \( 1 - 1.30T + 59T^{2} \)
61 \( 1 + 14.9T + 61T^{2} \)
67 \( 1 + 0.0645T + 67T^{2} \)
71 \( 1 + 7.40T + 71T^{2} \)
73 \( 1 + 4.96T + 73T^{2} \)
79 \( 1 + 3.28T + 79T^{2} \)
83 \( 1 + 2.68T + 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 - 1.63T + 97T^{2} \)
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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

−7.77686676053488193170718066469, −7.18717936564815332703700014059, −6.29616148505411027294667830819, −5.68803422783645702863347468513, −4.82966851032642784787673112373, −4.37861601224262722313147140622, −2.66074373342866920017279402354, −2.19008934239021513189002799799, −1.35700629050175259863854802370, 0, 1.35700629050175259863854802370, 2.19008934239021513189002799799, 2.66074373342866920017279402354, 4.37861601224262722313147140622, 4.82966851032642784787673112373, 5.68803422783645702863347468513, 6.29616148505411027294667830819, 7.18717936564815332703700014059, 7.77686676053488193170718066469

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