Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·7-s + 6·11-s − 3·13-s − 2·17-s + 19-s + 2·23-s + 6·29-s − 3·31-s − 6·37-s − 4·41-s − 11·43-s + 10·47-s + 18·49-s + 8·53-s + 6·59-s − 3·61-s + 67-s − 12·71-s − 10·73-s − 30·77-s + 8·79-s − 6·83-s + 16·89-s + 15·91-s + 7·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.88·7-s + 1.80·11-s − 0.832·13-s − 0.485·17-s + 0.229·19-s + 0.417·23-s + 1.11·29-s − 0.538·31-s − 0.986·37-s − 0.624·41-s − 1.67·43-s + 1.45·47-s + 18/7·49-s + 1.09·53-s + 0.781·59-s − 0.384·61-s + 0.122·67-s − 1.42·71-s − 1.17·73-s − 3.41·77-s + 0.900·79-s − 0.658·83-s + 1.69·89-s + 1.57·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-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  =  1
Selberg data  =  $(2,\ 14400,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 5 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−16.57331534191019, −15.86882823488458, −15.28471121043194, −14.78760368593971, −14.11081813221903, −13.53996389013075, −13.09761069934404, −12.30364344048662, −12.03440247679834, −11.52515506832375, −10.45265324340785, −10.11335111801188, −9.515620673410624, −8.896776235862387, −8.683844165925605, −7.339848972618305, −6.957038235278886, −6.499859976717686, −5.933740340253252, −5.055674538872839, −4.206677618580275, −3.571903628622055, −3.034084626244738, −2.131742662598131, −1.027846113192283, 0, 1.027846113192283, 2.131742662598131, 3.034084626244738, 3.571903628622055, 4.206677618580275, 5.055674538872839, 5.933740340253252, 6.499859976717686, 6.957038235278886, 7.339848972618305, 8.683844165925605, 8.896776235862387, 9.515620673410624, 10.11335111801188, 10.45265324340785, 11.52515506832375, 12.03440247679834, 12.30364344048662, 13.09761069934404, 13.53996389013075, 14.11081813221903, 14.78760368593971, 15.28471121043194, 15.86882823488458, 16.57331534191019

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