Properties

Degree 2
Conductor $ 2^{4} \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 + 4·7-s + 2·13-s − 6·17-s + 4·19-s + 25-s + 6·29-s − 8·31-s + 4·35-s + 2·37-s + 6·41-s + 4·43-s + 9·49-s + 6·53-s − 10·61-s + 2·65-s + 4·67-s + 2·73-s − 8·79-s + 12·83-s − 6·85-s − 18·89-s + 8·91-s + 4·95-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 9/7·49-s + 0.824·53-s − 1.28·61-s + 0.248·65-s + 0.488·67-s + 0.234·73-s − 0.900·79-s + 1.31·83-s − 0.650·85-s − 1.90·89-s + 0.838·91-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{720} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 720,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.935248229\)
\(L(\frac12)\)  \(\approx\)  \(1.935248229\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−19.69632559399246, −18.50001347708142, −17.91045392306185, −17.66113001614779, −16.65472146851745, −15.81967588144078, −15.07664792466368, −14.24783799342201, −13.75720498217786, −12.89856915909525, −11.88881174433111, −11.10193834894168, −10.67015888052156, −9.415401762585890, −8.713350404322888, −7.882979071679055, −6.961705597551280, −5.852363287733144, −4.970267860360122, −4.084714313611694, −2.503749623581634, −1.362437123647018, 1.362437123647018, 2.503749623581634, 4.084714313611694, 4.970267860360122, 5.852363287733144, 6.961705597551280, 7.882979071679055, 8.713350404322888, 9.415401762585890, 10.67015888052156, 11.10193834894168, 11.88881174433111, 12.89856915909525, 13.75720498217786, 14.24783799342201, 15.07664792466368, 15.81967588144078, 16.65472146851745, 17.66113001614779, 17.91045392306185, 18.50001347708142, 19.69632559399246

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