Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5·7-s + 9-s − 3·11-s + 13-s + 17-s + 7·19-s + 5·21-s − 6·23-s − 27-s + 9·29-s − 6·31-s + 3·33-s + 9·37-s − 39-s + 3·41-s − 6·43-s + 9·47-s + 18·49-s − 51-s − 3·53-s − 7·57-s + 14·59-s − 5·63-s − 14·67-s + 6·69-s + 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.88·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.242·17-s + 1.60·19-s + 1.09·21-s − 1.25·23-s − 0.192·27-s + 1.67·29-s − 1.07·31-s + 0.522·33-s + 1.47·37-s − 0.160·39-s + 0.468·41-s − 0.914·43-s + 1.31·47-s + 18/7·49-s − 0.140·51-s − 0.412·53-s − 0.927·57-s + 1.82·59-s − 0.629·63-s − 1.71·67-s + 0.722·69-s + 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(265200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{265200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 265200,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
13 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.05142230128396, −12.42843740773133, −12.23500049654580, −11.77667251116438, −11.17443166864106, −10.63424005861144, −10.16486910347425, −9.888211820855384, −9.463499998213407, −9.056952979811193, −8.212787903482277, −7.934978459010282, −7.167404813639146, −6.987114148611618, −6.303187759083532, −5.899118976820894, −5.540490964083668, −5.052876874988611, −4.192788478213056, −3.875238409705675, −3.139827069581755, −2.825230913619680, −2.243514847057453, −1.217741820423597, −0.6612757218116480, 0, 0.6612757218116480, 1.217741820423597, 2.243514847057453, 2.825230913619680, 3.139827069581755, 3.875238409705675, 4.192788478213056, 5.052876874988611, 5.540490964083668, 5.899118976820894, 6.303187759083532, 6.987114148611618, 7.167404813639146, 7.934978459010282, 8.212787903482277, 9.056952979811193, 9.463499998213407, 9.888211820855384, 10.16486910347425, 10.63424005861144, 11.17443166864106, 11.77667251116438, 12.23500049654580, 12.42843740773133, 13.05142230128396

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