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 − 18·101-s + 4·103-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 − 1.79·101-s + 0.394·103-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  :  $\chi_{720} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 720,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.93524\)
\(L(\frac12)\)  \(\approx\)  \(1.93524\)
\(L(\frac{3}{2})\)   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

−10.67015888052155601125139797767, −9.415401762585890013045544643092, −8.713350404322888410914105544855, −7.88297907167905508023143292047, −6.96170559755127999142057961261, −5.85236328773314391596499497567, −4.97026786036012165020386374880, −4.08471431361169401081275954745, −2.50374962358163435732612030839, −1.36243712364701817507861978801, 1.36243712364701817507861978801, 2.50374962358163435732612030839, 4.08471431361169401081275954745, 4.97026786036012165020386374880, 5.85236328773314391596499497567, 6.96170559755127999142057961261, 7.88297907167905508023143292047, 8.713350404322888410914105544855, 9.415401762585890013045544643092, 10.67015888052155601125139797767

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