Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 11 $
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 − 11-s + 12-s + 6·13-s − 14-s + 2·15-s − 16-s + 2·17-s − 18-s + 4·19-s + 2·20-s − 21-s + 22-s − 3·24-s − 25-s − 6·26-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 − 0.301·11-s + 0.288·12-s + 1.66·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.218·21-s + 0.213·22-s − 0.612·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(231\)    =    \(3 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{231} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 231,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5822888619$
$L(\frac12)$  $\approx$  $0.5822888619$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−19.68847492932172, −18.93323166431548, −18.15660522336797, −17.77188151688711, −16.53465606954257, −16.08298497435639, −15.11229263157731, −13.85480810785340, −13.14703415485481, −11.89946826402132, −11.18319190967267, −10.32896954095525, −9.238375783455942, −8.146329212122428, −7.613864008176004, −6.065071205390999, −4.780336543423643, −3.694786335913866, −1.020084495628171, 1.020084495628171, 3.694786335913866, 4.780336543423643, 6.065071205390999, 7.613864008176004, 8.146329212122428, 9.238375783455942, 10.32896954095525, 11.18319190967267, 11.89946826402132, 13.14703415485481, 13.85480810785340, 15.11229263157731, 16.08298497435639, 16.53465606954257, 17.77188151688711, 18.15660522336797, 18.93323166431548, 19.68847492932172

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