Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 17^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 4·11-s + 12-s + 2·13-s − 16-s + 18-s + 4·19-s + 4·22-s + 3·24-s + 2·26-s − 27-s + 2·29-s + 5·32-s − 4·33-s − 36-s − 10·37-s + 4·38-s − 2·39-s − 10·41-s − 4·43-s − 4·44-s − 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.852·22-s + 0.612·24-s + 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.883·32-s − 0.696·33-s − 1/6·36-s − 1.64·37-s + 0.648·38-s − 0.320·39-s − 1.56·41-s − 0.609·43-s − 0.603·44-s − 1.16·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21675\)    =    \(3 \cdot 5^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{21675} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 21675,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
17 \( 1 \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.79808733652950, −15.15916494515028, −14.72198242693331, −14.03431847450384, −13.70478976870578, −13.22250086926941, −12.50490113665343, −11.99474381433688, −11.69061024327973, −11.10801141090013, −10.24252832832036, −9.810857423433633, −9.168696354992937, −8.609806086799086, −8.104875420158932, −7.027064678144183, −6.666718117023006, −6.036755439123625, −5.362021528899911, −4.915930571037800, −4.222894055903174, −3.509425130450013, −3.180749275302384, −1.809693898840095, −1.081384441976826, 0, 1.081384441976826, 1.809693898840095, 3.180749275302384, 3.509425130450013, 4.222894055903174, 4.915930571037800, 5.362021528899911, 6.036755439123625, 6.666718117023006, 7.027064678144183, 8.104875420158932, 8.609806086799086, 9.168696354992937, 9.810857423433633, 10.24252832832036, 11.10801141090013, 11.69061024327973, 11.99474381433688, 12.50490113665343, 13.22250086926941, 13.70478976870578, 14.03431847450384, 14.72198242693331, 15.15916494515028, 15.79808733652950

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