Properties

Degree 2
Conductor $ 3^{2} \cdot 5 $
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 − 4-s − 5-s − 3·8-s − 10-s + 4·11-s − 2·13-s − 16-s − 2·17-s + 4·19-s + 20-s + 4·22-s + 25-s − 2·26-s + 2·29-s + 5·32-s − 2·34-s − 10·37-s + 4·38-s + 3·40-s − 10·41-s + 4·43-s − 4·44-s − 8·47-s − 7·49-s + 50-s + 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s + 1.20·11-s − 0.554·13-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 1/5·25-s − 0.392·26-s + 0.371·29-s + 0.883·32-s − 0.342·34-s − 1.64·37-s + 0.648·38-s + 0.474·40-s − 1.56·41-s + 0.609·43-s − 0.603·44-s − 1.16·47-s − 49-s + 0.141·50-s + 0.277·52-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(45\)    =    \(3^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{45} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 45,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9215908766$
$L(\frac12)$  $\approx$  $0.9215908766$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
5 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
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

−19.20380852088268, −18.00767588804966, −17.00262719846346, −15.60334701825816, −14.55874952579683, −13.71457429111567, −12.40658486355793, −11.57318950916086, −9.738851148469479, −8.554588868470700, −6.769558851589123, −5.048600900936683, −3.634128182367313, 3.634128182367313, 5.048600900936683, 6.769558851589123, 8.554588868470700, 9.738851148469479, 11.57318950916086, 12.40658486355793, 13.71457429111567, 14.55874952579683, 15.60334701825816, 17.00262719846346, 18.00767588804966, 19.20380852088268

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