Properties

Degree 2
Conductor $ 7 \cdot 23 $
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 − 4-s + 2·5-s + 7-s + 3·8-s − 3·9-s − 2·10-s + 4·11-s + 6·13-s − 14-s − 16-s − 2·17-s + 3·18-s + 4·19-s − 2·20-s − 4·22-s − 23-s − 25-s − 6·26-s − 28-s − 2·29-s − 4·31-s − 5·32-s + 2·34-s + 2·35-s + 3·36-s − 2·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 9-s − 0.632·10-s + 1.20·11-s + 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.707·18-s + 0.917·19-s − 0.447·20-s − 0.852·22-s − 0.208·23-s − 1/5·25-s − 1.17·26-s − 0.188·28-s − 0.371·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s + 1/2·36-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(161\)    =    \(7 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{161} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 161,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8628343267$
$L(\frac12)$  $\approx$  $0.8628343267$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−19.05424614766282, −18.07459393753372, −17.61331970279539, −16.90899938311434, −15.95093890967040, −14.38896442506988, −13.94988844245329, −13.12082677402823, −11.57306814136850, −10.82999474885335, −9.519763597988292, −8.955336427994799, −8.048209611049889, −6.434028060302365, −5.382003985683284, −3.764964445116421, −1.519268548540304, 1.519268548540304, 3.764964445116421, 5.382003985683284, 6.434028060302365, 8.048209611049889, 8.955336427994799, 9.519763597988292, 10.82999474885335, 11.57306814136850, 13.12082677402823, 13.94988844245329, 14.38896442506988, 15.95093890967040, 16.90899938311434, 17.61331970279539, 18.07459393753372, 19.05424614766282

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