Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 11^{2} $
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-s + 3-s − 4-s − 2·5-s + 6-s + 7-s − 3·8-s + 9-s − 2·10-s − 12-s + 2·13-s + 14-s − 2·15-s − 16-s + 6·17-s + 18-s − 4·19-s + 2·20-s + 21-s − 3·24-s − 25-s + 2·26-s + 27-s − 28-s + 2·29-s − 2·30-s + 5·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.365·30-s + 0.883·32-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  =  0
Selberg data  =  $(2,\ 2541,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.304740393$
$L(\frac12)$  $\approx$  $2.304740393$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
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

−19.09596762821552, −18.37611824954840, −17.84486582336623, −16.93940671605389, −16.25565148507381, −15.47199013207948, −14.94760107428179, −14.45416546547210, −13.84332799687643, −13.15796858012131, −12.49597014375472, −11.93911102596347, −11.23727948227840, −10.30302436393845, −9.563733444203339, −8.699165629902587, −8.166185648107227, −7.598605305113226, −6.501047952237606, −5.627673766991474, −4.791352170722338, −3.918052774078995, −3.565996843964124, −2.434703228304793, −0.8605694561736051, 0.8605694561736051, 2.434703228304793, 3.565996843964124, 3.918052774078995, 4.791352170722338, 5.627673766991474, 6.501047952237606, 7.598605305113226, 8.166185648107227, 8.699165629902587, 9.563733444203339, 10.30302436393845, 11.23727948227840, 11.93911102596347, 12.49597014375472, 13.15796858012131, 13.84332799687643, 14.45416546547210, 14.94760107428179, 15.47199013207948, 16.25565148507381, 16.93940671605389, 17.84486582336623, 18.37611824954840, 19.09596762821552

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