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 − 2·37-s − 39-s + 6·41-s − 4·43-s + 9·49-s − 51-s − 6·53-s + 4·57-s + 12·59-s − 10·61-s − 4·63-s − 4·67-s + 12·71-s + 10·73-s + 16·79-s + 81-s − 12·83-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 − 0.328·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.140·51-s − 0.824·53-s + 0.529·57-s + 1.56·59-s − 1.28·61-s − 0.503·63-s − 0.488·67-s + 1.42·71-s + 1.17·73-s + 1.80·79-s + 1/9·81-s − 1.31·83-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 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.95606453443816, −12.59140154601310, −12.29319920825298, −11.72261955846481, −11.20976126146946, −10.57998564439811, −10.08966692634309, −9.809645608109435, −9.314237948000611, −9.024163313820363, −8.327694947955147, −7.954859425358183, −7.374702300426302, −6.851526896713223, −6.475946164051824, −6.090716667567603, −5.337437336586504, −4.885744899304970, −4.256211333629781, −3.602486709758963, −3.318442293706513, −2.609923120158469, −2.420630077156261, −1.392767232373586, −0.7886842502999189, 0, 0.7886842502999189, 1.392767232373586, 2.420630077156261, 2.609923120158469, 3.318442293706513, 3.602486709758963, 4.256211333629781, 4.885744899304970, 5.337437336586504, 6.090716667567603, 6.475946164051824, 6.851526896713223, 7.374702300426302, 7.954859425358183, 8.327694947955147, 9.024163313820363, 9.314237948000611, 9.809645608109435, 10.08966692634309, 10.57998564439811, 11.20976126146946, 11.72261955846481, 12.29319920825298, 12.59140154601310, 12.95606453443816

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