Properties

Degree 2
Conductor $ 3^{2} \cdot 19 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 3·5-s − 5·7-s + 6·10-s − 11-s + 2·13-s − 10·14-s − 4·16-s + 17-s − 19-s + 6·20-s − 2·22-s + 4·23-s + 4·25-s + 4·26-s − 10·28-s + 2·29-s − 6·31-s − 8·32-s + 2·34-s − 15·35-s − 2·38-s − 43-s − 2·44-s + 8·46-s + 9·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.34·5-s − 1.88·7-s + 1.89·10-s − 0.301·11-s + 0.554·13-s − 2.67·14-s − 16-s + 0.242·17-s − 0.229·19-s + 1.34·20-s − 0.426·22-s + 0.834·23-s + 4/5·25-s + 0.784·26-s − 1.88·28-s + 0.371·29-s − 1.07·31-s − 1.41·32-s + 0.342·34-s − 2.53·35-s − 0.324·38-s − 0.152·43-s − 0.301·44-s + 1.17·46-s + 1.31·47-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  =  0
Selberg data  =  $(2,\ 171,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.215096110$
$L(\frac12)$  $\approx$  $2.215096110$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−19.08722396653137, −18.25872549309349, −17.10761908277375, −16.18692452404747, −15.43185307035999, −14.30288646563613, −13.48809242040455, −13.00980511664591, −12.37438605696104, −10.84385583714429, −9.769325574936890, −9.028755790922139, −6.850738863502884, −6.122417516098507, −5.346681462526589, −3.678407967813732, −2.616879983815507, 2.616879983815507, 3.678407967813732, 5.346681462526589, 6.122417516098507, 6.850738863502884, 9.028755790922139, 9.769325574936890, 10.84385583714429, 12.37438605696104, 13.00980511664591, 13.48809242040455, 14.30288646563613, 15.43185307035999, 16.18692452404747, 17.10761908277375, 18.25872549309349, 19.08722396653137

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