Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \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  + 5-s + 2·7-s + 2·13-s + 6·17-s − 4·19-s − 6·23-s + 25-s − 6·29-s − 4·31-s + 2·35-s + 2·37-s − 6·41-s − 10·43-s + 6·47-s − 3·49-s + 6·53-s − 12·59-s + 2·61-s + 2·65-s + 2·67-s + 12·71-s + 2·73-s + 8·79-s − 6·83-s + 6·85-s + 6·89-s + 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.338·35-s + 0.328·37-s − 0.937·41-s − 1.52·43-s + 0.875·47-s − 3/7·49-s + 0.824·53-s − 1.56·59-s + 0.256·61-s + 0.248·65-s + 0.244·67-s + 1.42·71-s + 0.234·73-s + 0.900·79-s − 0.658·83-s + 0.650·85-s + 0.635·89-s + 0.419·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{180} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 180,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.312989889$
$L(\frac12)$  $\approx$  $1.312989889$
$L(\frac{3}{2})$   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 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−19.89119695423677, −18.61503695321336, −18.25232108752574, −17.07718151760702, −16.53999275930739, −15.27632744445510, −14.48531421733238, −13.69863664669158, −12.64919926214782, −11.66835822739294, −10.66869217043212, −9.749524176059453, −8.536040373596851, −7.645897782815644, −6.214441157829271, −5.196478485344312, −3.711197934118167, −1.817930152520926, 1.817930152520926, 3.711197934118167, 5.196478485344312, 6.214441157829271, 7.645897782815644, 8.536040373596851, 9.749524176059453, 10.66869217043212, 11.66835822739294, 12.64919926214782, 13.69863664669158, 14.48531421733238, 15.27632744445510, 16.53999275930739, 17.07718151760702, 18.25232108752574, 18.61503695321336, 19.89119695423677

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