Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.69·2-s + 3-s + 5.24·4-s + 3.40·5-s − 2.69·6-s − 8.74·8-s + 9-s − 9.17·10-s + 2.47·11-s + 5.24·12-s − 2.61·13-s + 3.40·15-s + 13.0·16-s − 5.06·17-s − 2.69·18-s − 2.94·19-s + 17.8·20-s − 6.67·22-s + 2.48·23-s − 8.74·24-s + 6.61·25-s + 7.05·26-s + 27-s + 0.267·29-s − 9.17·30-s − 6.04·31-s − 17.6·32-s + ⋯
L(s)  = 1  − 1.90·2-s + 0.577·3-s + 2.62·4-s + 1.52·5-s − 1.09·6-s − 3.09·8-s + 0.333·9-s − 2.90·10-s + 0.746·11-s + 1.51·12-s − 0.726·13-s + 0.880·15-s + 3.26·16-s − 1.22·17-s − 0.634·18-s − 0.675·19-s + 4.00·20-s − 1.42·22-s + 0.517·23-s − 1.78·24-s + 1.32·25-s + 1.38·26-s + 0.192·27-s + 0.0496·29-s − 1.67·30-s − 1.08·31-s − 3.12·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6027} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.381171973$
$L(\frac12)$  $\approx$  $1.381171973$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 + 2.69T + 2T^{2} \)
5 \( 1 - 3.40T + 5T^{2} \)
11 \( 1 - 2.47T + 11T^{2} \)
13 \( 1 + 2.61T + 13T^{2} \)
17 \( 1 + 5.06T + 17T^{2} \)
19 \( 1 + 2.94T + 19T^{2} \)
23 \( 1 - 2.48T + 23T^{2} \)
29 \( 1 - 0.267T + 29T^{2} \)
31 \( 1 + 6.04T + 31T^{2} \)
37 \( 1 - 9.78T + 37T^{2} \)
43 \( 1 - 7.25T + 43T^{2} \)
47 \( 1 + 8.30T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + 2.98T + 59T^{2} \)
61 \( 1 + 7.07T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 - 8.21T + 71T^{2} \)
73 \( 1 + 6.66T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + 7.72T + 97T^{2} \)
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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

−8.318368747581158407588958925676, −7.50187730770427081567990388115, −6.78970055531667773415847413971, −6.38271861612146278163769273343, −5.59316756057547422098910886950, −4.40820551285955347719469261595, −3.07099900995286324335740319858, −2.14975499794910886929938777139, −1.96296214338983068212924027870, −0.791038749591541721280906337176, 0.791038749591541721280906337176, 1.96296214338983068212924027870, 2.14975499794910886929938777139, 3.07099900995286324335740319858, 4.40820551285955347719469261595, 5.59316756057547422098910886950, 6.38271861612146278163769273343, 6.78970055531667773415847413971, 7.50187730770427081567990388115, 8.318368747581158407588958925676

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