Properties

Degree 2
Conductor $ 3 \cdot 7 \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 + 3-s − 4-s + 2·5-s + 6-s + 7-s − 3·8-s + 9-s + 2·10-s − 4·11-s − 12-s + 14-s + 2·15-s − 16-s − 6·17-s + 18-s − 4·19-s − 2·20-s + 21-s − 4·22-s − 3·24-s − 25-s + 27-s − 28-s − 2·29-s + 2·30-s + 5·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.288·12-s + 0.267·14-s + 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s − 0.852·22-s − 0.612·24-s − 1/5·25-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.365·30-s + 0.883·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{3549} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 3549,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−18.41893296232032, −18.14343335094381, −17.46643734061182, −16.97411804451301, −15.84544061443214, −15.32165575910822, −14.85886858452822, −14.07017850990880, −13.54594347531838, −13.22614456678341, −12.67005807146321, −11.84054109861693, −10.85220679895575, −10.31355171347969, −9.542092147710695, −8.829118090640207, −8.407377139641224, −7.487283801655556, −6.506447479227431, −5.819341899526437, −5.005852134911911, −4.481666301491806, −3.519546433683499, −2.531336089482615, −1.884535722732817, 0, 1.884535722732817, 2.531336089482615, 3.519546433683499, 4.481666301491806, 5.005852134911911, 5.819341899526437, 6.506447479227431, 7.487283801655556, 8.407377139641224, 8.829118090640207, 9.542092147710695, 10.31355171347969, 10.85220679895575, 11.84054109861693, 12.67005807146321, 13.22614456678341, 13.54594347531838, 14.07017850990880, 14.85886858452822, 15.32165575910822, 15.84544061443214, 16.97411804451301, 17.46643734061182, 18.14343335094381, 18.41893296232032

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