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 − 4·7-s + 9-s − 13-s − 17-s − 4·19-s + 4·21-s − 27-s + 6·29-s − 4·31-s + 6·37-s + 39-s − 2·41-s − 4·43-s + 9·49-s + 51-s + 10·53-s + 4·57-s + 4·59-s + 14·61-s − 4·63-s − 12·67-s + 12·71-s − 14·73-s + 81-s + 12·83-s − 6·87-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.242·17-s − 0.917·19-s + 0.872·21-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.986·37-s + 0.160·39-s − 0.312·41-s − 0.609·43-s + 9/7·49-s + 0.140·51-s + 1.37·53-s + 0.529·57-s + 0.520·59-s + 1.79·61-s − 0.503·63-s − 1.46·67-s + 1.42·71-s − 1.63·73-s + 1/9·81-s + 1.31·83-s − 0.643·87-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−12.96462996843714, −12.58881541151819, −12.18588699412602, −11.65819988394871, −11.26778682117704, −10.57135499217027, −10.27904272943284, −9.901509991290576, −9.413581342597420, −8.877228162221019, −8.484324715666375, −7.841463292144780, −7.198324598619096, −6.810976311918604, −6.418960330428932, −6.025178505566492, −5.454920943691808, −4.932808550481021, −4.290176126965818, −3.843037176656435, −3.321742817344838, −2.580680671927785, −2.278170240887312, −1.320901878743350, −0.5944722364532790, 0, 0.5944722364532790, 1.320901878743350, 2.278170240887312, 2.580680671927785, 3.321742817344838, 3.843037176656435, 4.290176126965818, 4.932808550481021, 5.454920943691808, 6.025178505566492, 6.418960330428932, 6.810976311918604, 7.198324598619096, 7.841463292144780, 8.484324715666375, 8.877228162221019, 9.413581342597420, 9.901509991290576, 10.27904272943284, 10.57135499217027, 11.26778682117704, 11.65819988394871, 12.18588699412602, 12.58881541151819, 12.96462996843714

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