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  − 4·11-s − 2·13-s + 2·17-s − 4·19-s − 2·29-s − 10·37-s − 10·41-s + 4·43-s + 8·47-s − 7·49-s + 10·53-s − 4·59-s + 2·61-s + 12·67-s + 8·71-s − 10·73-s − 12·83-s + 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.371·29-s − 1.64·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 49-s + 1.37·53-s − 0.520·59-s + 0.256·61-s + 1.46·67-s + 0.949·71-s − 1.17·73-s − 1.31·83-s + 0.635·89-s − 0.203·97-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  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 14400,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.165730496$
$L(\frac12)$  $\approx$  $1.165730496$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
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

−16.03750093923812, −15.44424398926866, −15.20080237614087, −14.33622802806449, −13.99727130809963, −13.16306019151609, −12.84435687325235, −12.19918203493078, −11.67244052241415, −10.89316347622021, −10.33433218868170, −10.04739886587379, −9.182952791722422, −8.528796002193558, −8.052642541080547, −7.302657719474862, −6.871265386233794, −5.967443688898475, −5.327908368970966, −4.862752361111817, −3.974861985736359, −3.235645399760779, −2.434627616398996, −1.769533749676322, −0.4539499407622705, 0.4539499407622705, 1.769533749676322, 2.434627616398996, 3.235645399760779, 3.974861985736359, 4.862752361111817, 5.327908368970966, 5.967443688898475, 6.871265386233794, 7.302657719474862, 8.052642541080547, 8.528796002193558, 9.182952791722422, 10.04739886587379, 10.33433218868170, 10.89316347622021, 11.67244052241415, 12.19918203493078, 12.84435687325235, 13.16306019151609, 13.99727130809963, 14.33622802806449, 15.20080237614087, 15.44424398926866, 16.03750093923812

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