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 − 3·7-s + 9-s + 2·11-s + 13-s − 17-s − 6·19-s + 3·21-s + 4·23-s − 27-s + 3·29-s − 2·33-s + 37-s − 39-s + 43-s − 9·47-s + 2·49-s + 51-s + 6·57-s − 4·59-s − 10·61-s − 3·63-s − 14·67-s − 4·69-s + 6·71-s − 14·73-s − 6·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.242·17-s − 1.37·19-s + 0.654·21-s + 0.834·23-s − 0.192·27-s + 0.557·29-s − 0.348·33-s + 0.164·37-s − 0.160·39-s + 0.152·43-s − 1.31·47-s + 2/7·49-s + 0.140·51-s + 0.794·57-s − 0.520·59-s − 1.28·61-s − 0.377·63-s − 1.71·67-s − 0.481·69-s + 0.712·71-s − 1.63·73-s − 0.683·77-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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−13.04291408120953, −12.61237820509340, −12.07904782101065, −11.74644447683481, −11.12175612769670, −10.71528367950586, −10.35839897257942, −9.822279019775001, −9.307436922249273, −8.937624439914861, −8.500919177216087, −7.828852828694054, −7.278458707871958, −6.712819132606401, −6.341739530413254, −6.157065157412437, −5.493573323655385, −4.739757699445470, −4.442785685598990, −3.855779094025031, −3.176899700921741, −2.874331866111006, −1.971225781061247, −1.442452356592023, −0.6307040949703546, 0, 0.6307040949703546, 1.442452356592023, 1.971225781061247, 2.874331866111006, 3.176899700921741, 3.855779094025031, 4.442785685598990, 4.739757699445470, 5.493573323655385, 6.157065157412437, 6.341739530413254, 6.712819132606401, 7.278458707871958, 7.828852828694054, 8.500919177216087, 8.937624439914861, 9.307436922249273, 9.822279019775001, 10.35839897257942, 10.71528367950586, 11.12175612769670, 11.74644447683481, 12.07904782101065, 12.61237820509340, 13.04291408120953

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