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 + 5·13-s + 2·14-s + 16-s − 3·17-s − 4·19-s − 20-s + 3·22-s − 9·23-s + 25-s + 5·26-s + 2·28-s − 3·29-s + 5·31-s + 32-s − 3·34-s − 2·35-s − 10·37-s − 4·38-s − 40-s − 43-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 + 1.38·13-s + 0.534·14-s + 1/4·16-s − 0.727·17-s − 0.917·19-s − 0.223·20-s + 0.639·22-s − 1.87·23-s + 1/5·25-s + 0.980·26-s + 0.377·28-s − 0.557·29-s + 0.898·31-s + 0.176·32-s − 0.514·34-s − 0.338·35-s − 1.64·37-s − 0.648·38-s − 0.158·40-s − 0.152·43-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  =  $(2,\ 270,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.93997$
$L(\frac12)$  $\approx$  $1.93997$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−11.82445088836287841705889069139, −11.26604299806316376931516007678, −10.33316270206535335896037451217, −8.800517236152472581876501034748, −8.091714346845311378630476626834, −6.76212554562659462902022059457, −5.89935357898068173165036223225, −4.44860777449897097026241543960, −3.71657663450820725474165684624, −1.82136551723493939659256186424, 1.82136551723493939659256186424, 3.71657663450820725474165684624, 4.44860777449897097026241543960, 5.89935357898068173165036223225, 6.76212554562659462902022059457, 8.091714346845311378630476626834, 8.800517236152472581876501034748, 10.33316270206535335896037451217, 11.26604299806316376931516007678, 11.82445088836287841705889069139

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