Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s − 3·5-s + 6-s − 7-s + 3·8-s + 9-s + 3·10-s + 12-s − 7·13-s + 14-s + 3·15-s − 16-s − 3·17-s − 18-s + 2·19-s + 3·20-s + 21-s − 4·23-s − 3·24-s + 4·25-s + 7·26-s − 27-s + 28-s − 7·29-s − 3·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.948·10-s + 0.288·12-s − 1.94·13-s + 0.267·14-s + 0.774·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.458·19-s + 0.670·20-s + 0.218·21-s − 0.834·23-s − 0.612·24-s + 4/5·25-s + 1.37·26-s − 0.192·27-s + 0.188·28-s − 1.29·29-s − 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2541} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 2541,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−19.43677481881269, −19.02335818703943, −18.18742513233868, −17.81700792711442, −16.98064834033625, −16.46519438650579, −16.07000415234933, −15.03693255604926, −14.74003583008022, −13.70352913674225, −12.87254966064703, −12.36299568990922, −11.67741667973306, −11.02035375341848, −10.29346151762470, −9.493565444849538, −9.095101571494047, −7.923421319655960, −7.541891311876948, −6.995285039846881, −5.727915631093539, −4.794287917701023, −4.266881387878708, −3.318102708391646, −1.841183518115588, 0, 0, 1.841183518115588, 3.318102708391646, 4.266881387878708, 4.794287917701023, 5.727915631093539, 6.995285039846881, 7.541891311876948, 7.923421319655960, 9.095101571494047, 9.493565444849538, 10.29346151762470, 11.02035375341848, 11.67741667973306, 12.36299568990922, 12.87254966064703, 13.70352913674225, 14.74003583008022, 15.03693255604926, 16.07000415234933, 16.46519438650579, 16.98064834033625, 17.81700792711442, 18.18742513233868, 19.02335818703943, 19.43677481881269

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