Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 $
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 − 7-s + 13-s − 6·17-s + 4·19-s + 25-s − 6·29-s + 4·31-s + 35-s − 10·37-s − 6·41-s − 8·43-s + 49-s + 6·53-s − 12·59-s + 14·61-s − 65-s + 4·67-s + 2·73-s − 8·79-s − 12·83-s + 6·85-s − 6·89-s − 91-s − 4·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 0.277·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.169·35-s − 1.64·37-s − 0.937·41-s − 1.21·43-s + 1/7·49-s + 0.824·53-s − 1.56·59-s + 1.79·61-s − 0.124·65-s + 0.488·67-s + 0.234·73-s − 0.900·79-s − 1.31·83-s + 0.650·85-s − 0.635·89-s − 0.104·91-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(65520\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{65520} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 65520,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8308439049$
$L(\frac12)$  $\approx$  $0.8308439049$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
good11 \( 1 + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
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

−14.08815506204505, −13.69764035257235, −13.20238038875878, −12.81333730986090, −12.11313365083516, −11.66384292773365, −11.28479628413838, −10.71821035719240, −10.09981432508260, −9.714365566499359, −8.934713307896969, −8.673240074000746, −8.090281775824563, −7.405469632899435, −6.820481206456425, −6.621839018711982, −5.719667122664242, −5.239601570257307, −4.619174804304723, −3.938927033327076, −3.455832288971565, −2.837097188171638, −2.034071020833544, −1.360630310567443, −0.3068042759442141, 0.3068042759442141, 1.360630310567443, 2.034071020833544, 2.837097188171638, 3.455832288971565, 3.938927033327076, 4.619174804304723, 5.239601570257307, 5.719667122664242, 6.621839018711982, 6.820481206456425, 7.405469632899435, 8.090281775824563, 8.673240074000746, 8.934713307896969, 9.714365566499359, 10.09981432508260, 10.71821035719240, 11.28479628413838, 11.66384292773365, 12.11313365083516, 12.81333730986090, 13.20238038875878, 13.69764035257235, 14.08815506204505

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