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  =  $(2,\ 262080,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.847473079$
$L(\frac12)$  $\approx$  $1.847473079$
$L(\frac{3}{2})$   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

−12.93100656363422, −12.27400691174051, −11.92979184452805, −11.35631982479860, −10.99184255483078, −10.59523890398426, −9.931472737344707, −9.610965085404620, −9.109577004934785, −8.682079823148718, −8.119593781445270, −7.687137483163295, −7.154234585270823, −6.564825258205109, −6.274480306356216, −5.661814892538358, −5.019000178746653, −4.690576024779001, −4.255421774994684, −3.434187961828801, −2.925753267169030, −2.404771686211166, −1.717069566671117, −1.287940865244988, −0.3612041185884374, 0.3612041185884374, 1.287940865244988, 1.717069566671117, 2.404771686211166, 2.925753267169030, 3.434187961828801, 4.255421774994684, 4.690576024779001, 5.019000178746653, 5.661814892538358, 6.274480306356216, 6.564825258205109, 7.154234585270823, 7.687137483163295, 8.119593781445270, 8.682079823148718, 9.109577004934785, 9.610965085404620, 9.931472737344707, 10.59523890398426, 10.99184255483078, 11.35631982479860, 11.92979184452805, 12.27400691174051, 12.93100656363422

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