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 − 2·7-s + 9-s − 4·11-s − 13-s − 17-s − 6·19-s + 2·21-s + 8·23-s − 27-s + 6·29-s + 8·31-s + 4·33-s + 2·37-s + 39-s − 2·41-s − 10·43-s − 4·47-s − 3·49-s + 51-s + 2·53-s + 6·57-s + 14·59-s − 10·61-s − 2·63-s + 12·67-s − 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.242·17-s − 1.37·19-s + 0.436·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.160·39-s − 0.312·41-s − 1.52·43-s − 0.583·47-s − 3/7·49-s + 0.140·51-s + 0.274·53-s + 0.794·57-s + 1.82·59-s − 1.28·61-s − 0.251·63-s + 1.46·67-s − 0.963·69-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−12.89149884368283, −12.73272589220000, −12.17149913684048, −11.61302019701396, −11.12094905966910, −10.78537319091946, −10.11010442345842, −10.03064573481622, −9.509089925027143, −8.667681573874535, −8.421727285822796, −8.010887285066569, −7.165348617853432, −6.864803329757206, −6.434618225491900, −6.027793693051681, −5.262903143244612, −4.906026315782800, −4.558235775635974, −3.830915656333933, −3.128660352961745, −2.710184664392117, −2.224519779987872, −1.339082700700702, −0.6119253532084352, 0, 0.6119253532084352, 1.339082700700702, 2.224519779987872, 2.710184664392117, 3.128660352961745, 3.830915656333933, 4.558235775635974, 4.906026315782800, 5.262903143244612, 6.027793693051681, 6.434618225491900, 6.864803329757206, 7.165348617853432, 8.010887285066569, 8.421727285822796, 8.667681573874535, 9.509089925027143, 10.03064573481622, 10.11010442345842, 10.78537319091946, 11.12094905966910, 11.61302019701396, 12.17149913684048, 12.73272589220000, 12.89149884368283

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