Properties

Degree 2
Conductor $ 2 \cdot 7^{2} \cdot 13^{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 + 2·3-s + 4-s + 2·6-s + 8-s + 9-s + 2·12-s + 16-s − 6·17-s + 18-s + 2·19-s + 2·24-s − 5·25-s − 4·27-s − 6·29-s − 4·31-s + 32-s − 6·34-s + 36-s − 2·37-s + 2·38-s + 6·41-s + 8·43-s − 12·47-s + 2·48-s − 5·50-s − 12·51-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.577·12-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.458·19-s + 0.408·24-s − 25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 0.288·48-s − 0.707·50-s − 1.68·51-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16562 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16562 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16562\)    =    \(2 \cdot 7^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{16562} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 16562,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 \)
13 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.93203997356574, −15.37465798001230, −15.15155158131758, −14.35924873858919, −14.06738861581008, −13.50184435655665, −13.01184363650562, −12.59010441071026, −11.66594883268868, −11.27618244937599, −10.72295879176170, −9.863444761110308, −9.215857103171911, −8.945193235668955, −8.040500744654836, −7.637914566721434, −7.016812070547293, −6.231754803897045, −5.639242089597259, −4.879015750862173, −4.042314247301108, −3.681492479977720, −2.825638740604823, −2.241881172581246, −1.581062460810565, 0, 1.581062460810565, 2.241881172581246, 2.825638740604823, 3.681492479977720, 4.042314247301108, 4.879015750862173, 5.639242089597259, 6.231754803897045, 7.016812070547293, 7.637914566721434, 8.040500744654836, 8.945193235668955, 9.215857103171911, 9.863444761110308, 10.72295879176170, 11.27618244937599, 11.66594883268868, 12.59010441071026, 13.01184363650562, 13.50184435655665, 14.06738861581008, 14.35924873858919, 15.15155158131758, 15.37465798001230, 15.93203997356574

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