Properties

Degree 2
Conductor $ 3 \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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 − 2·13-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 + 2·26-s + 27-s + 28-s − 2·29-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 − 0.554·13-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 + 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{21} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 21,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4511154053$
$L(\frac12)$  $\approx$  $0.4511154053$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
7 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
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

−19.89701463571632, −19.66784881755091, −18.41922826796405, −17.15982418199832, −15.87345980006495, −14.51353234037879, −13.19379250527464, −11.55595081754560, −9.724661210726174, −8.672365196683145, −7.250477838028384, −4.135590840507737, 4.135590840507737, 7.250477838028384, 8.672365196683145, 9.724661210726174, 11.55595081754560, 13.19379250527464, 14.51353234037879, 15.87345980006495, 17.15982418199832, 18.41922826796405, 19.66784881755091, 19.89701463571632

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