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  − 1.63·2-s + 3-s + 0.673·4-s + 1.18·5-s − 1.63·6-s + 2.16·8-s + 9-s − 1.94·10-s + 6.03·11-s + 0.673·12-s − 6.76·13-s + 1.18·15-s − 4.89·16-s + 2.16·17-s − 1.63·18-s − 0.190·19-s + 0.799·20-s − 9.87·22-s + 1.56·23-s + 2.16·24-s − 3.59·25-s + 11.0·26-s + 27-s − 1.86·29-s − 1.94·30-s − 4.59·31-s + 3.66·32-s + ⋯
L(s)  = 1  − 1.15·2-s + 0.577·3-s + 0.336·4-s + 0.530·5-s − 0.667·6-s + 0.766·8-s + 0.333·9-s − 0.613·10-s + 1.82·11-s + 0.194·12-s − 1.87·13-s + 0.306·15-s − 1.22·16-s + 0.525·17-s − 0.385·18-s − 0.0437·19-s + 0.178·20-s − 2.10·22-s + 0.326·23-s + 0.442·24-s − 0.718·25-s + 2.16·26-s + 0.192·27-s − 0.346·29-s − 0.354·30-s − 0.826·31-s + 0.647·32-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 \( 1 + 1.63T + 2T^{2} \)
5 \( 1 - 1.18T + 5T^{2} \)
11 \( 1 - 6.03T + 11T^{2} \)
13 \( 1 + 6.76T + 13T^{2} \)
17 \( 1 - 2.16T + 17T^{2} \)
19 \( 1 + 0.190T + 19T^{2} \)
23 \( 1 - 1.56T + 23T^{2} \)
29 \( 1 + 1.86T + 29T^{2} \)
31 \( 1 + 4.59T + 31T^{2} \)
37 \( 1 + 9.75T + 37T^{2} \)
43 \( 1 + 3.27T + 43T^{2} \)
47 \( 1 - 7.29T + 47T^{2} \)
53 \( 1 + 8.09T + 53T^{2} \)
59 \( 1 + 6.53T + 59T^{2} \)
61 \( 1 + 2.90T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 - 1.12T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 6.13T + 83T^{2} \)
89 \( 1 - 8.62T + 89T^{2} \)
97 \( 1 + 6.82T + 97T^{2} \)
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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

−7.64229104471781442061297180045, −7.33905667818515114950391755233, −6.64431535496992369875155075943, −5.65657830311037584410575016845, −4.73220396482848370621260258890, −4.03726343454935486928493219940, −3.04063840191267017731676480679, −1.91341257445156413278608766476, −1.44043103471221921736478963248, 0, 1.44043103471221921736478963248, 1.91341257445156413278608766476, 3.04063840191267017731676480679, 4.03726343454935486928493219940, 4.73220396482848370621260258890, 5.65657830311037584410575016845, 6.64431535496992369875155075943, 7.33905667818515114950391755233, 7.64229104471781442061297180045

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