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 + 8·19-s + 2·21-s − 27-s + 2·29-s + 6·31-s + 4·33-s + 4·37-s + 39-s + 2·41-s − 4·43-s − 6·47-s − 3·49-s + 51-s − 6·53-s − 8·57-s + 10·59-s − 10·61-s − 2·63-s + 4·67-s − 8·71-s + 8·77-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 + 1.83·19-s + 0.436·21-s − 0.192·27-s + 0.371·29-s + 1.07·31-s + 0.696·33-s + 0.657·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.875·47-s − 3/7·49-s + 0.140·51-s − 0.824·53-s − 1.05·57-s + 1.30·59-s − 1.28·61-s − 0.251·63-s + 0.488·67-s − 0.949·71-s + 0.911·77-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−13.10102380590468, −12.48266157862505, −12.16188049234151, −11.56210211564442, −11.35946100562892, −10.63469197549600, −10.20078851830600, −9.942627744555140, −9.381809931040366, −9.050412839695097, −8.146890059472607, −7.882778859600338, −7.430404797780364, −6.856439020843058, −6.303996711156572, −6.035119546843011, −5.234315308480814, −5.010218703207252, −4.563669262516269, −3.702867175746536, −3.202281396032343, −2.769922595769844, −2.171156854010432, −1.295966585368103, −0.6822293684216237, 0, 0.6822293684216237, 1.295966585368103, 2.171156854010432, 2.769922595769844, 3.202281396032343, 3.702867175746536, 4.563669262516269, 5.010218703207252, 5.234315308480814, 6.035119546843011, 6.303996711156572, 6.856439020843058, 7.430404797780364, 7.882778859600338, 8.146890059472607, 9.050412839695097, 9.381809931040366, 9.942627744555140, 10.20078851830600, 10.63469197549600, 11.35946100562892, 11.56210211564442, 12.16188049234151, 12.48266157862505, 13.10102380590468

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