Properties

Degree 2
Conductor $ 3^{2} \cdot 19 $
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 − 3·5-s − 7-s − 3·11-s − 4·13-s + 4·16-s + 3·17-s + 19-s + 6·20-s + 4·25-s + 2·28-s − 6·29-s − 4·31-s + 3·35-s + 2·37-s + 6·41-s − 43-s + 6·44-s + 3·47-s − 6·49-s + 8·52-s − 12·53-s + 9·55-s + 6·59-s − 61-s − 8·64-s + 12·65-s + ⋯
L(s)  = 1  − 4-s − 1.34·5-s − 0.377·7-s − 0.904·11-s − 1.10·13-s + 16-s + 0.727·17-s + 0.229·19-s + 1.34·20-s + 4/5·25-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.507·35-s + 0.328·37-s + 0.937·41-s − 0.152·43-s + 0.904·44-s + 0.437·47-s − 6/7·49-s + 1.10·52-s − 1.64·53-s + 1.21·55-s + 0.781·59-s − 0.128·61-s − 64-s + 1.48·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(171\)    =    \(3^{2} \cdot 19\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{171} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 171,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.27070724267868, −18.80547242361029, −17.87165368053570, −16.80738761180438, −15.99361654061062, −15.01989762648940, −14.29678548933660, −12.99225309233174, −12.45987383937425, −11.37237044309941, −10.15333178148615, −9.238767264318382, −7.968751144610877, −7.431414033120004, −5.510979052837361, −4.414662066511427, −3.238575521735675, 0, 3.238575521735675, 4.414662066511427, 5.510979052837361, 7.431414033120004, 7.968751144610877, 9.238767264318382, 10.15333178148615, 11.37237044309941, 12.45987383937425, 12.99225309233174, 14.29678548933660, 15.01989762648940, 15.99361654061062, 16.80738761180438, 17.87165368053570, 18.80547242361029, 19.27070724267868

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