Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s − 6·5-s − 16·7-s + 8·8-s − 12·10-s − 12·11-s + 38·13-s − 32·14-s + 16·16-s + 126·17-s + 20·19-s − 24·20-s − 24·22-s − 168·23-s − 89·25-s + 76·26-s − 64·28-s − 30·29-s − 88·31-s + 32·32-s + 252·34-s + 96·35-s + 254·37-s + 40·38-s − 48·40-s − 42·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.536·5-s − 0.863·7-s + 0.353·8-s − 0.379·10-s − 0.328·11-s + 0.810·13-s − 0.610·14-s + 1/4·16-s + 1.79·17-s + 0.241·19-s − 0.268·20-s − 0.232·22-s − 1.52·23-s − 0.711·25-s + 0.573·26-s − 0.431·28-s − 0.192·29-s − 0.509·31-s + 0.176·32-s + 1.27·34-s + 0.463·35-s + 1.12·37-s + 0.170·38-s − 0.189·40-s − 0.159·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(18\)    =    \(2 \cdot 3^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{18} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 18,\ (\ :3/2),\ 1)$
$L(2)$  $\approx$  $1.33993$
$L(\frac12)$  $\approx$  $1.33993$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - p T \)
3 \( 1 \)
good5 \( 1 + 6 T + p^{3} T^{2} \)
7 \( 1 + 16 T + p^{3} T^{2} \)
11 \( 1 + 12 T + p^{3} T^{2} \)
13 \( 1 - 38 T + p^{3} T^{2} \)
17 \( 1 - 126 T + p^{3} T^{2} \)
19 \( 1 - 20 T + p^{3} T^{2} \)
23 \( 1 + 168 T + p^{3} T^{2} \)
29 \( 1 + 30 T + p^{3} T^{2} \)
31 \( 1 + 88 T + p^{3} T^{2} \)
37 \( 1 - 254 T + p^{3} T^{2} \)
41 \( 1 + 42 T + p^{3} T^{2} \)
43 \( 1 + 52 T + p^{3} T^{2} \)
47 \( 1 - 96 T + p^{3} T^{2} \)
53 \( 1 + 198 T + p^{3} T^{2} \)
59 \( 1 - 660 T + p^{3} T^{2} \)
61 \( 1 + 538 T + p^{3} T^{2} \)
67 \( 1 - 884 T + p^{3} T^{2} \)
71 \( 1 + 792 T + p^{3} T^{2} \)
73 \( 1 - 218 T + p^{3} T^{2} \)
79 \( 1 + 520 T + p^{3} T^{2} \)
83 \( 1 - 492 T + p^{3} T^{2} \)
89 \( 1 + 810 T + p^{3} T^{2} \)
97 \( 1 - 1154 T + p^{3} T^{2} \)
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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

−18.44187078213435321181228660570, −16.49283416198810241904062500247, −15.69789007701018280405867310968, −14.21619922753104885303434400485, −12.88299997955728899305459192822, −11.69135476711267912580710906154, −9.997177907882694374396051296820, −7.79784374777438716958608823195, −5.91841366397310679443986977087, −3.60929026962551384976945217111, 3.60929026962551384976945217111, 5.91841366397310679443986977087, 7.79784374777438716958608823195, 9.997177907882694374396051296820, 11.69135476711267912580710906154, 12.88299997955728899305459192822, 14.21619922753104885303434400485, 15.69789007701018280405867310968, 16.49283416198810241904062500247, 18.44187078213435321181228660570

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