Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 7^{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 + 4-s + 8-s + 4·13-s + 16-s + 6·17-s − 2·19-s − 5·25-s + 4·26-s + 6·29-s + 4·31-s + 32-s + 6·34-s + 2·37-s − 2·38-s + 6·41-s + 8·43-s − 12·47-s − 5·50-s + 4·52-s − 6·53-s + 6·58-s − 6·59-s − 8·61-s + 4·62-s + 64-s − 4·67-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.458·19-s − 25-s + 0.784·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s − 0.324·38-s + 0.937·41-s + 1.21·43-s − 1.75·47-s − 0.707·50-s + 0.554·52-s − 0.824·53-s + 0.787·58-s − 0.781·59-s − 1.02·61-s + 0.508·62-s + 1/8·64-s − 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{882} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 882,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.59418$
$L(\frac12)$  $\approx$  $2.59418$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−10.27339907123230780461643050534, −9.393998538973797562285424645673, −8.245010328271296459357780366393, −7.64010214491199898083268392854, −6.37349457460642388728354325328, −5.88407913223931048395780461484, −4.74683922157615295493069282209, −3.79391663707115759548696099652, −2.83833445427180997893704571146, −1.33827299261372059142982522511, 1.33827299261372059142982522511, 2.83833445427180997893704571146, 3.79391663707115759548696099652, 4.74683922157615295493069282209, 5.88407913223931048395780461484, 6.37349457460642388728354325328, 7.64010214491199898083268392854, 8.245010328271296459357780366393, 9.393998538973797562285424645673, 10.27339907123230780461643050534

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