Properties

Degree 2
Conductor $ 3 \cdot 11 \cdot 47^{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 + 4·7-s − 3·8-s + 9-s + 2·10-s − 11-s + 12-s + 2·13-s + 4·14-s − 2·15-s − 16-s − 2·17-s + 18-s − 2·20-s − 4·21-s − 22-s − 8·23-s + 3·24-s − 25-s + 2·26-s − 27-s − 4·28-s + 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.447·20-s − 0.872·21-s − 0.213·22-s − 1.66·23-s + 0.612·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 72897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 72897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(72897\)    =    \(3 \cdot 11 \cdot 47^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{72897} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 72897,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.603113292$
$L(\frac12)$  $\approx$  $3.603113292$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;47\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;47\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
11 \( 1 + T \)
47 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.01664064445573, −13.70270520345324, −13.25875300717129, −12.61100756576642, −12.13203496359075, −11.61764615858015, −11.29333178443116, −10.58355868430938, −10.06963715892632, −9.724658641102546, −8.989868642678213, −8.313592810535341, −8.166393830908442, −7.440520162594419, −6.460728632517970, −6.143118470703632, −5.733576812887000, −5.022756828682224, −4.747656124846562, −4.179206189975345, −3.641210576130372, −2.522623642993669, −2.183198931434465, −1.287443851725668, −0.6220188627890586, 0.6220188627890586, 1.287443851725668, 2.183198931434465, 2.522623642993669, 3.641210576130372, 4.179206189975345, 4.747656124846562, 5.022756828682224, 5.733576812887000, 6.143118470703632, 6.460728632517970, 7.440520162594419, 8.166393830908442, 8.313592810535341, 8.989868642678213, 9.724658641102546, 10.06963715892632, 10.58355868430938, 11.29333178443116, 11.61764615858015, 12.13203496359075, 12.61100756576642, 13.25875300717129, 13.70270520345324, 14.01664064445573

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