Properties

Degree 2
Conductor $ 2^{6} \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

\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(262080\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{262080} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 262080,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.762486077$
$L(\frac12)$  $\approx$  $1.762486077$
$L(\frac{3}{2})$   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

−12.79202978234483, −12.48680283201989, −11.89134484458698, −11.38863339008802, −10.95879678148811, −10.45190761832427, −10.07842947854473, −9.595881490127419, −8.971570915498561, −8.824423654025805, −8.121951096343844, −7.736847331084327, −6.991182316135058, −6.624389154099080, −6.188006031823276, −5.857495401070644, −5.025028818742705, −4.533607135670691, −4.287047603626694, −3.525452787787334, −2.752720461556326, −2.495516392858442, −1.886396444098597, −1.104575608029370, −0.3771227312713436, 0.3771227312713436, 1.104575608029370, 1.886396444098597, 2.495516392858442, 2.752720461556326, 3.525452787787334, 4.287047603626694, 4.533607135670691, 5.025028818742705, 5.857495401070644, 6.188006031823276, 6.624389154099080, 6.991182316135058, 7.736847331084327, 8.121951096343844, 8.824423654025805, 8.971570915498561, 9.595881490127419, 10.07842947854473, 10.45190761832427, 10.95879678148811, 11.38863339008802, 11.89134484458698, 12.48680283201989, 12.79202978234483

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