Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{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  + 4·7-s + 2·13-s − 8·19-s − 5·25-s + 4·31-s − 10·37-s − 8·43-s + 9·49-s + 14·61-s + 16·67-s − 10·73-s + 4·79-s + 8·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.51·7-s + 0.554·13-s − 1.83·19-s − 25-s + 0.718·31-s − 1.64·37-s − 1.21·43-s + 9/7·49-s + 1.79·61-s + 1.95·67-s − 1.17·73-s + 0.450·79-s + 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(144\)    =    \(2^{4} \cdot 3^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{144} (1, \cdot )$
Sato-Tate  :  $N(\mathrm{U}(1))$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 144,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.214325323$
$L(\frac12)$  $\approx$  $1.214325323$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−19.43353618498370, −18.60771093678513, −17.57217856336291, −17.14416929609552, −15.79487777633049, −14.94637601970120, −14.14961250099870, −13.16759329932275, −11.94687294246885, −11.12315991811672, −10.22525649903538, −8.662053039688616, −8.061333242868227, −6.626967388340823, −5.240606646992834, −4.050614281322311, −1.922099012735744, 1.922099012735744, 4.050614281322311, 5.240606646992834, 6.626967388340823, 8.061333242868227, 8.662053039688616, 10.22525649903538, 11.12315991811672, 11.94687294246885, 13.16759329932275, 14.14961250099870, 14.94637601970120, 15.79487777633049, 17.14416929609552, 17.57217856336291, 18.60771093678513, 19.43353618498370

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