Properties

Degree 2
Conductor $ 3^{2} $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 4·4-s − 6·5-s − 40·7-s − 168·8-s − 36·10-s + 564·11-s + 638·13-s − 240·14-s − 1.13e3·16-s − 882·17-s − 556·19-s − 24·20-s + 3.38e3·22-s + 840·23-s − 3.08e3·25-s + 3.82e3·26-s − 160·28-s − 4.63e3·29-s + 4.40e3·31-s − 1.44e3·32-s − 5.29e3·34-s + 240·35-s − 2.41e3·37-s − 3.33e3·38-s + 1.00e3·40-s + 6.87e3·41-s + ⋯
L(s)  = 1  + 1.06·2-s + 1/8·4-s − 0.107·5-s − 0.308·7-s − 0.928·8-s − 0.113·10-s + 1.40·11-s + 1.04·13-s − 0.327·14-s − 1.10·16-s − 0.740·17-s − 0.353·19-s − 0.0134·20-s + 1.49·22-s + 0.331·23-s − 0.988·25-s + 1.11·26-s − 0.0385·28-s − 1.02·29-s + 0.822·31-s − 0.248·32-s − 0.785·34-s + 0.0331·35-s − 0.289·37-s − 0.374·38-s + 0.0996·40-s + 0.638·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9\)    =    \(3^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{9} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 9,\ (\ :5/2),\ 1)$
$L(3)$  $\approx$  $1.62095$
$L(\frac12)$  $\approx$  $1.62095$
$L(\frac{7}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
good2 \( 1 - 3 p T + p^{5} T^{2} \)
5 \( 1 + 6 T + p^{5} T^{2} \)
7 \( 1 + 40 T + p^{5} T^{2} \)
11 \( 1 - 564 T + p^{5} T^{2} \)
13 \( 1 - 638 T + p^{5} T^{2} \)
17 \( 1 + 882 T + p^{5} T^{2} \)
19 \( 1 + 556 T + p^{5} T^{2} \)
23 \( 1 - 840 T + p^{5} T^{2} \)
29 \( 1 + 4638 T + p^{5} T^{2} \)
31 \( 1 - 4400 T + p^{5} T^{2} \)
37 \( 1 + 2410 T + p^{5} T^{2} \)
41 \( 1 - 6870 T + p^{5} T^{2} \)
43 \( 1 - 9644 T + p^{5} T^{2} \)
47 \( 1 - 18672 T + p^{5} T^{2} \)
53 \( 1 + 33750 T + p^{5} T^{2} \)
59 \( 1 - 18084 T + p^{5} T^{2} \)
61 \( 1 - 39758 T + p^{5} T^{2} \)
67 \( 1 + 23068 T + p^{5} T^{2} \)
71 \( 1 - 4248 T + p^{5} T^{2} \)
73 \( 1 + 41110 T + p^{5} T^{2} \)
79 \( 1 - 21920 T + p^{5} T^{2} \)
83 \( 1 + 82452 T + p^{5} T^{2} \)
89 \( 1 - 94086 T + p^{5} T^{2} \)
97 \( 1 - 49442 T + p^{5} T^{2} \)
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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

−20.72885339876540857140390360645, −19.16593764231050914774156590454, −17.49468626833473430194637999306, −15.67933525354395910079610816331, −14.27824140496176556494038470951, −13.04070313597022456551357571710, −11.52275722912076873690315140301, −9.071675575237721781933448979598, −6.25120604714416436509917537703, −3.96221921058525041818495599724, 3.96221921058525041818495599724, 6.25120604714416436509917537703, 9.071675575237721781933448979598, 11.52275722912076873690315140301, 13.04070313597022456551357571710, 14.27824140496176556494038470951, 15.67933525354395910079610816331, 17.49468626833473430194637999306, 19.16593764231050914774156590454, 20.72885339876540857140390360645

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