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 2

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 + 2·19-s + 2·21-s − 27-s + 2·29-s + 4·33-s + 2·37-s + 39-s + 2·41-s + 6·43-s − 12·47-s − 3·49-s − 51-s − 10·53-s − 2·57-s − 6·59-s − 10·61-s − 2·63-s − 4·67-s − 10·73-s + 8·77-s − 6·79-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 + 0.458·19-s + 0.436·21-s − 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.328·37-s + 0.160·39-s + 0.312·41-s + 0.914·43-s − 1.75·47-s − 3/7·49-s − 0.140·51-s − 1.37·53-s − 0.264·57-s − 0.781·59-s − 1.28·61-s − 0.251·63-s − 0.488·67-s − 1.17·73-s + 0.911·77-s − 0.675·79-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  =  2
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 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.13176070907272, −12.65026505082138, −12.53611286139429, −11.91454671565162, −11.31756870636183, −11.03153210440499, −10.46000329659030, −10.03868187727930, −9.659895013575301, −9.266896732786146, −8.581063854610289, −7.989928915705255, −7.652172617409509, −7.161183927335830, −6.596462966411158, −6.107009653951108, −5.689045189954831, −5.207331049243017, −4.566984209896652, −4.331833384750431, −3.259830518147101, −3.108086135292631, −2.506069757732195, −1.677445754413313, −1.094017273593708, 0, 0, 1.094017273593708, 1.677445754413313, 2.506069757732195, 3.108086135292631, 3.259830518147101, 4.331833384750431, 4.566984209896652, 5.207331049243017, 5.689045189954831, 6.107009653951108, 6.596462966411158, 7.161183927335830, 7.652172617409509, 7.989928915705255, 8.581063854610289, 9.266896732786146, 9.659895013575301, 10.03868187727930, 10.46000329659030, 11.03153210440499, 11.31756870636183, 11.91454671565162, 12.53611286139429, 12.65026505082138, 13.13176070907272

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