Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 13^{2} $
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-s − 4-s − 2·5-s + 7-s + 3·8-s + 2·10-s + 4·11-s − 14-s − 16-s + 6·17-s − 4·19-s + 2·20-s − 4·22-s − 25-s − 28-s + 2·29-s − 5·32-s − 6·34-s − 2·35-s − 6·37-s + 4·38-s − 6·40-s + 2·41-s − 4·43-s − 4·44-s + 49-s + 50-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s + 1.06·8-s + 0.632·10-s + 1.20·11-s − 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.188·28-s + 0.371·29-s − 0.883·32-s − 1.02·34-s − 0.338·35-s − 0.986·37-s + 0.648·38-s − 0.948·40-s + 0.312·41-s − 0.609·43-s − 0.603·44-s + 1/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10647 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10647 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10647\)    =    \(3^{2} \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{10647} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 10647,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.89846285312187, −16.48313072459138, −15.78736386477614, −15.11189985910555, −14.47407120252318, −14.16355188329994, −13.49473949441358, −12.55338040178547, −12.29227884657985, −11.46764922089658, −11.12900757168578, −10.16707194084633, −9.890744284220179, −9.076691393972949, −8.423786399769656, −8.188368053984681, −7.370695544008836, −6.898482559938685, −5.919186307157493, −5.164830411161612, −4.295106664237148, −3.949078440449165, −3.135455550611383, −1.764364863804771, −1.074849960182110, 0, 1.074849960182110, 1.764364863804771, 3.135455550611383, 3.949078440449165, 4.295106664237148, 5.164830411161612, 5.919186307157493, 6.898482559938685, 7.370695544008836, 8.188368053984681, 8.423786399769656, 9.076691393972949, 9.890744284220179, 10.16707194084633, 11.12900757168578, 11.46764922089658, 12.29227884657985, 12.55338040178547, 13.49473949441358, 14.16355188329994, 14.47407120252318, 15.11189985910555, 15.78736386477614, 16.48313072459138, 16.89846285312187

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