Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \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 − 4-s + 5-s − 3·8-s + 10-s − 4·11-s − 2·13-s − 16-s + 4·19-s − 20-s − 4·22-s + 25-s − 2·26-s − 2·29-s + 5·32-s + 10·37-s + 4·38-s − 3·40-s + 10·41-s + 4·43-s + 4·44-s − 8·47-s − 7·49-s + 50-s + 2·52-s + 10·53-s − 4·55-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s + 0.316·10-s − 1.20·11-s − 0.554·13-s − 1/4·16-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 1/5·25-s − 0.392·26-s − 0.371·29-s + 0.883·32-s + 1.64·37-s + 0.648·38-s − 0.474·40-s + 1.56·41-s + 0.609·43-s + 0.603·44-s − 1.16·47-s − 49-s + 0.141·50-s + 0.277·52-s + 1.37·53-s − 0.539·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13005\)    =    \(3^{2} \cdot 5 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{13005} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 13005,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
5 \( 1 - T \)
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

−16.25794422916592, −16.08417834225541, −15.24277135805808, −14.61370026886122, −14.41046746010312, −13.59992679475216, −13.10502701845389, −12.88202619301199, −12.15929435906700, −11.49977957062198, −10.90857422240175, −10.05884084020756, −9.659032377141479, −9.169993140191653, −8.295304542255877, −7.788173656511871, −7.097700418103183, −6.202442201487989, −5.571393723692607, −5.198788611400258, −4.484821033109937, −3.809232481771646, −2.792686980916850, −2.517718348346171, −1.120177446235366, 0, 1.120177446235366, 2.517718348346171, 2.792686980916850, 3.809232481771646, 4.484821033109937, 5.198788611400258, 5.571393723692607, 6.202442201487989, 7.097700418103183, 7.788173656511871, 8.295304542255877, 9.169993140191653, 9.659032377141479, 10.05884084020756, 10.90857422240175, 11.49977957062198, 12.15929435906700, 12.88202619301199, 13.10502701845389, 13.59992679475216, 14.41046746010312, 14.61370026886122, 15.24277135805808, 16.08417834225541, 16.25794422916592

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