Properties

Degree 2
Conductor 83
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 − 3·7-s + 3·8-s − 2·9-s + 2·10-s + 3·11-s + 12-s − 6·13-s + 3·14-s + 2·15-s − 16-s + 5·17-s + 2·18-s + 2·19-s + 2·20-s + 3·21-s − 3·22-s − 4·23-s − 3·24-s − 25-s + 6·26-s + 5·27-s + 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.13·7-s + 1.06·8-s − 2/3·9-s + 0.632·10-s + 0.904·11-s + 0.288·12-s − 1.66·13-s + 0.801·14-s + 0.516·15-s − 1/4·16-s + 1.21·17-s + 0.471·18-s + 0.458·19-s + 0.447·20-s + 0.654·21-s − 0.639·22-s − 0.834·23-s − 0.612·24-s − 1/5·25-s + 1.17·26-s + 0.962·27-s + 0.566·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{83} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 83,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 83$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 83$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad83 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−19.42783350289885, −18.98433131920491, −17.62714815703440, −16.85569317903090, −16.30410363710809, −14.86720481310773, −13.82349410001976, −12.30534590659786, −11.77890434965261, −10.17676635448238, −9.457827669916619, −8.118927993931558, −6.974502993044303, −5.313270241803154, −3.654759012444532, 0, 3.654759012444532, 5.313270241803154, 6.974502993044303, 8.118927993931558, 9.457827669916619, 10.17676635448238, 11.77890434965261, 12.30534590659786, 13.82349410001976, 14.86720481310773, 16.30410363710809, 16.85569317903090, 17.62714815703440, 18.98433131920491, 19.42783350289885

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