Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s + 2·13-s + 8·19-s + 4·31-s − 10·37-s − 8·43-s + 9·49-s − 14·61-s + 16·67-s + 10·73-s + 4·79-s − 8·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.51·7-s + 0.554·13-s + 1.83·19-s + 0.718·31-s − 1.64·37-s − 1.21·43-s + 9/7·49-s − 1.79·61-s + 1.95·67-s + 1.17·73-s + 0.450·79-s − 0.838·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(14400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{14400} (1, \cdot )$
Sato-Tate  :  $N(\mathrm{U}(1))$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 14400,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.536013537$
$L(\frac12)$  $\approx$  $1.536013537$
$L(\frac{3}{2})$   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 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.01299107280611, −15.63470346663305, −15.27742358387992, −14.26110654525624, −13.72755911294391, −13.48111681182649, −12.70541547510739, −12.21947594918532, −11.70722253370840, −10.98801693887082, −10.29294160095649, −9.765779400741374, −9.376028204622992, −8.675080332563171, −8.002676936916516, −7.220744126266057, −6.706451804819759, −6.154252036149070, −5.430748765469350, −4.823817108703231, −3.657956575312888, −3.400277674411254, −2.670529205753197, −1.537271731484684, −0.5567078400165889, 0.5567078400165889, 1.537271731484684, 2.670529205753197, 3.400277674411254, 3.657956575312888, 4.823817108703231, 5.430748765469350, 6.154252036149070, 6.706451804819759, 7.220744126266057, 8.002676936916516, 8.675080332563171, 9.376028204622992, 9.765779400741374, 10.29294160095649, 10.98801693887082, 11.70722253370840, 12.21947594918532, 12.70541547510739, 13.48111681182649, 13.72755911294391, 14.26110654525624, 15.27742358387992, 15.63470346663305, 16.01299107280611

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