Properties

Degree 2
Conductor 163
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 4·5-s + 2·7-s − 3·9-s − 6·11-s + 4·13-s + 4·16-s − 6·19-s + 8·20-s + 6·23-s + 11·25-s − 4·28-s − 4·29-s − 6·31-s − 8·35-s + 6·36-s − 8·37-s + 3·41-s + 7·43-s + 12·44-s + 12·45-s + 47-s − 3·49-s − 8·52-s − 9·53-s + 24·55-s − 2·59-s + ⋯
L(s)  = 1  − 4-s − 1.78·5-s + 0.755·7-s − 9-s − 1.80·11-s + 1.10·13-s + 16-s − 1.37·19-s + 1.78·20-s + 1.25·23-s + 11/5·25-s − 0.755·28-s − 0.742·29-s − 1.07·31-s − 1.35·35-s + 36-s − 1.31·37-s + 0.468·41-s + 1.06·43-s + 1.80·44-s + 1.78·45-s + 0.145·47-s − 3/7·49-s − 1.10·52-s − 1.23·53-s + 3.23·55-s − 0.260·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(163\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{163} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 163,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 163$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 163$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad163 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−19.32830286431667, −18.82937373066191, −18.02007569010231, −17.04237477873111, −15.96225742695998, −15.14725600291372, −14.42877801478807, −13.18008073542967, −12.48626235146584, −10.99632573201438, −10.92315118072157, −8.786931194290833, −8.324796826494764, −7.520910688089608, −5.496280773010057, −4.485762093603258, −3.291876576795656, 0, 3.291876576795656, 4.485762093603258, 5.496280773010057, 7.520910688089608, 8.324796826494764, 8.786931194290833, 10.92315118072157, 10.99632573201438, 12.48626235146584, 13.18008073542967, 14.42877801478807, 15.14725600291372, 15.96225742695998, 17.04237477873111, 18.02007569010231, 18.82937373066191, 19.32830286431667

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