Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.04·2-s + 3-s − 0.904·4-s + 1.81·5-s − 1.04·6-s + 3.03·8-s + 9-s − 1.89·10-s − 0.445·11-s − 0.904·12-s − 2.67·13-s + 1.81·15-s − 1.37·16-s + 2.86·17-s − 1.04·18-s − 4.73·19-s − 1.64·20-s + 0.466·22-s + 3.59·23-s + 3.03·24-s − 1.71·25-s + 2.79·26-s + 27-s − 4.22·29-s − 1.89·30-s + 6.94·31-s − 4.64·32-s + ⋯
L(s)  = 1  − 0.739·2-s + 0.577·3-s − 0.452·4-s + 0.810·5-s − 0.427·6-s + 1.07·8-s + 0.333·9-s − 0.600·10-s − 0.134·11-s − 0.261·12-s − 0.740·13-s + 0.468·15-s − 0.342·16-s + 0.695·17-s − 0.246·18-s − 1.08·19-s − 0.366·20-s + 0.0995·22-s + 0.750·23-s + 0.620·24-s − 0.342·25-s + 0.548·26-s + 0.192·27-s − 0.783·29-s − 0.346·30-s + 1.24·31-s − 0.821·32-s + ⋯

Functional equation

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

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  =  1
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 + 1.04T + 2T^{2} \)
5 \( 1 - 1.81T + 5T^{2} \)
11 \( 1 + 0.445T + 11T^{2} \)
13 \( 1 + 2.67T + 13T^{2} \)
17 \( 1 - 2.86T + 17T^{2} \)
19 \( 1 + 4.73T + 19T^{2} \)
23 \( 1 - 3.59T + 23T^{2} \)
29 \( 1 + 4.22T + 29T^{2} \)
31 \( 1 - 6.94T + 31T^{2} \)
37 \( 1 + 9.17T + 37T^{2} \)
43 \( 1 + 7.79T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 5.38T + 59T^{2} \)
61 \( 1 + 7.26T + 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 - 5.02T + 79T^{2} \)
83 \( 1 + 7.60T + 83T^{2} \)
89 \( 1 - 2.20T + 89T^{2} \)
97 \( 1 - 1.75T + 97T^{2} \)
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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

−7.927962744130487980948126744972, −7.19408360266569386234666667851, −6.49547081161596417665838953555, −5.46223028467164463531186534615, −4.88649276680690861240467868337, −4.04303642992105197852059978555, −3.08652163420045366981007194698, −2.10582699487173326214156979208, −1.38648133853996815355722447252, 0, 1.38648133853996815355722447252, 2.10582699487173326214156979208, 3.08652163420045366981007194698, 4.04303642992105197852059978555, 4.88649276680690861240467868337, 5.46223028467164463531186534615, 6.49547081161596417665838953555, 7.19408360266569386234666667851, 7.927962744130487980948126744972

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