Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \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  − 5-s − 4·7-s − 4·11-s − 2·13-s − 2·17-s + 4·19-s − 4·23-s + 25-s + 2·29-s − 8·31-s + 4·35-s + 6·37-s + 6·41-s − 8·43-s − 4·47-s + 9·49-s − 6·53-s + 4·55-s + 4·59-s − 2·61-s + 2·65-s + 8·67-s − 6·73-s + 16·77-s + 16·83-s + 2·85-s + 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.676·35-s + 0.986·37-s + 0.937·41-s − 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s + 0.520·59-s − 0.256·61-s + 0.248·65-s + 0.977·67-s − 0.702·73-s + 1.82·77-s + 1.75·83-s + 0.216·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{360} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 360,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−19.94405606080938, −19.33511862237906, −18.45666268636171, −17.86029007999850, −16.55104093187096, −16.09465684100002, −15.48308829516393, −14.47438770433399, −13.34128298874155, −12.84510311361010, −11.99387343035677, −10.92459683746192, −9.969026297182817, −9.341400856476565, −8.048390911320260, −7.218578105599622, −6.181780552180544, −5.058024276583353, −3.657886735203656, −2.617724659800455, 0, 2.617724659800455, 3.657886735203656, 5.058024276583353, 6.181780552180544, 7.218578105599622, 8.048390911320260, 9.341400856476565, 9.969026297182817, 10.92459683746192, 11.99387343035677, 12.84510311361010, 13.34128298874155, 14.47438770433399, 15.48308829516393, 16.09465684100002, 16.55104093187096, 17.86029007999850, 18.45666268636171, 19.33511862237906, 19.94405606080938

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