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 − 2·11-s + 13-s − 17-s − 2·19-s + 4·21-s − 8·23-s − 27-s + 2·29-s − 4·31-s + 2·33-s − 8·37-s − 39-s − 10·41-s − 8·47-s + 9·49-s + 51-s + 2·53-s + 2·57-s + 6·61-s − 4·63-s − 10·67-s + 8·69-s − 6·71-s + 16·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.242·17-s − 0.458·19-s + 0.872·21-s − 1.66·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.348·33-s − 1.31·37-s − 0.160·39-s − 1.56·41-s − 1.16·47-s + 9/7·49-s + 0.140·51-s + 0.274·53-s + 0.264·57-s + 0.768·61-s − 0.503·63-s − 1.22·67-s + 0.963·69-s − 0.712·71-s + 1.87·73-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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−12.88383694580428, −12.65845644845423, −12.09626371677660, −11.77933078848843, −11.18451699799139, −10.55846173846415, −10.29733445492415, −9.906198259213304, −9.474871854925427, −8.839410124144062, −8.410599547391081, −7.890251069788662, −7.279508472022738, −6.723766827508727, −6.427841742353945, −6.036970802988471, −5.359175466315305, −5.077447954973995, −4.261769306843553, −3.748142166607517, −3.379252145346189, −2.716160259829155, −2.045351227272351, −1.517133809354703, −0.4357469849486253, 0, 0.4357469849486253, 1.517133809354703, 2.045351227272351, 2.716160259829155, 3.379252145346189, 3.748142166607517, 4.261769306843553, 5.077447954973995, 5.359175466315305, 6.036970802988471, 6.427841742353945, 6.723766827508727, 7.279508472022738, 7.890251069788662, 8.410599547391081, 8.839410124144062, 9.474871854925427, 9.906198259213304, 10.29733445492415, 10.55846173846415, 11.18451699799139, 11.77933078848843, 12.09626371677660, 12.65845644845423, 12.88383694580428

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