Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 5-s − 3·7-s + 2·10-s − 2·11-s − 5·13-s + 6·14-s − 4·16-s − 8·17-s + 19-s − 2·20-s + 4·22-s + 6·23-s + 25-s + 10·26-s − 6·28-s + 2·29-s + 8·32-s + 16·34-s + 3·35-s + 5·37-s − 2·38-s − 10·41-s + 4·43-s − 4·44-s − 12·46-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s − 1.13·7-s + 0.632·10-s − 0.603·11-s − 1.38·13-s + 1.60·14-s − 16-s − 1.94·17-s + 0.229·19-s − 0.447·20-s + 0.852·22-s + 1.25·23-s + 1/5·25-s + 1.96·26-s − 1.13·28-s + 0.371·29-s + 1.41·32-s + 2.74·34-s + 0.507·35-s + 0.821·37-s − 0.324·38-s − 1.56·41-s + 0.609·43-s − 0.603·44-s − 1.76·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(135\)    =    \(3^{3} \cdot 5\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{135} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 135,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
5 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−19.71622153842706, −19.28517270407292, −18.40324124757740, −17.47481001768801, −16.73465612419349, −15.83283093068544, −15.15491017462618, −13.52240844513659, −12.66426764168114, −11.36787525907074, −10.40592213360929, −9.509466913311256, −8.705039129288460, −7.445840331358636, −6.687997253676356, −4.692216909383447, −2.652549933985925, 0, 2.652549933985925, 4.692216909383447, 6.687997253676356, 7.445840331358636, 8.705039129288460, 9.509466913311256, 10.40592213360929, 11.36787525907074, 12.66426764168114, 13.52240844513659, 15.15491017462618, 15.83283093068544, 16.73465612419349, 17.47481001768801, 18.40324124757740, 19.28517270407292, 19.71622153842706

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