Properties

Degree 2
Conductor $ 3^{2} \cdot 7^{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 − 3·8-s − 2·10-s − 4·11-s + 2·13-s − 16-s − 6·17-s − 4·19-s + 2·20-s − 4·22-s − 25-s + 2·26-s + 2·29-s + 5·32-s − 6·34-s + 6·37-s − 4·38-s + 6·40-s + 2·41-s − 4·43-s + 4·44-s − 50-s − 2·52-s − 6·53-s + 8·55-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s − 1.20·11-s + 0.554·13-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 + 0.392·26-s + 0.371·29-s + 0.883·32-s − 1.02·34-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 − 0.141·50-s − 0.277·52-s − 0.824·53-s + 1.07·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(441\)    =    \(3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{441} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 441,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
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

−19.54534563745202, −18.72195877055626, −18.07936206975170, −17.36012103458747, −16.10788625255945, −15.51752268278982, −14.95565340351815, −13.93343707235102, −13.09484579286367, −12.73191094564425, −11.57536344069931, −10.92437047522814, −9.800766905340478, −8.609463409035165, −8.123916451859490, −6.814228560272174, −5.732677116331321, −4.619143093685349, −3.943289176563945, −2.648397461386886, 0, 2.648397461386886, 3.943289176563945, 4.619143093685349, 5.732677116331321, 6.814228560272174, 8.123916451859490, 8.609463409035165, 9.800766905340478, 10.92437047522814, 11.57536344069931, 12.73191094564425, 13.09484579286367, 13.93343707235102, 14.95565340351815, 15.51752268278982, 16.10788625255945, 17.36012103458747, 18.07936206975170, 18.72195877055626, 19.54534563745202

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