Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 5 $
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 − 5-s + 2·7-s − 8-s + 10-s + 3·11-s − 13-s − 2·14-s + 16-s + 3·17-s + 8·19-s − 20-s − 3·22-s − 3·23-s + 25-s + 26-s + 2·28-s + 9·29-s − 7·31-s − 32-s − 3·34-s − 2·35-s + 2·37-s − 8·38-s + 40-s + 12·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.904·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s + 1.83·19-s − 0.223·20-s − 0.639·22-s − 0.625·23-s + 1/5·25-s + 0.196·26-s + 0.377·28-s + 1.67·29-s − 1.25·31-s − 0.176·32-s − 0.514·34-s − 0.338·35-s + 0.328·37-s − 1.29·38-s + 0.158·40-s + 1.87·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{270} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 270,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.987144$
$L(\frac12)$  $\approx$  $0.987144$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
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

−11.78318272116545337419358605651, −11.05191214604594379403959723854, −9.898970025925554421885060612926, −9.095398543889443614795538732083, −7.955240080686076886252892838851, −7.36179448350786937184598515633, −6.02650666217825993704928776127, −4.68795794788034775526265746527, −3.21325548378725257481862846236, −1.33830057852400417582257636134, 1.33830057852400417582257636134, 3.21325548378725257481862846236, 4.68795794788034775526265746527, 6.02650666217825993704928776127, 7.36179448350786937184598515633, 7.955240080686076886252892838851, 9.095398543889443614795538732083, 9.898970025925554421885060612926, 11.05191214604594379403959723854, 11.78318272116545337419358605651

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