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Properties

Label 2-171-19.18-c2-0-10
Degree $2$
Conductor $171$
Sign $-0.397 + 0.917i$
Analytic cond. $4.65941$
Root an. cond. $2.15856$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.04i·2-s − 5.27·4-s + 6.27·5-s + 12.2·7-s + 3.88i·8-s − 19.1i·10-s − 0.274·11-s + 13.0i·13-s − 37.3i·14-s − 9.27·16-s − 17.3·17-s + (7.54 − 17.4i)19-s − 33.0·20-s + 0.837i·22-s − 20.5·23-s + ⋯
L(s)  = 1  − 1.52i·2-s − 1.31·4-s + 1.25·5-s + 1.75·7-s + 0.485i·8-s − 1.91i·10-s − 0.0249·11-s + 1.00i·13-s − 2.67i·14-s − 0.579·16-s − 1.02·17-s + (0.397 − 0.917i)19-s − 1.65·20-s + 0.0380i·22-s − 0.893·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.397 + 0.917i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ -0.397 + 0.917i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05967 - 1.61360i\)
\(L(\frac12)\) \(\approx\) \(1.05967 - 1.61360i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (-7.54 + 17.4i)T \)
good2 \( 1 + 3.04iT - 4T^{2} \)
5 \( 1 - 6.27T + 25T^{2} \)
7 \( 1 - 12.2T + 49T^{2} \)
11 \( 1 + 0.274T + 121T^{2} \)
13 \( 1 - 13.0iT - 169T^{2} \)
17 \( 1 + 17.3T + 289T^{2} \)
23 \( 1 + 20.5T + 529T^{2} \)
29 \( 1 - 26.0iT - 841T^{2} \)
31 \( 1 - 23.5iT - 961T^{2} \)
37 \( 1 + 66.7iT - 1.36e3T^{2} \)
41 \( 1 - 3.57iT - 1.68e3T^{2} \)
43 \( 1 + 48.1T + 1.84e3T^{2} \)
47 \( 1 - 12.4T + 2.20e3T^{2} \)
53 \( 1 + 25.8iT - 2.80e3T^{2} \)
59 \( 1 - 0.230iT - 3.48e3T^{2} \)
61 \( 1 + 28.8T + 3.72e3T^{2} \)
67 \( 1 - 102. iT - 4.48e3T^{2} \)
71 \( 1 - 107. iT - 5.04e3T^{2} \)
73 \( 1 - 11.1T + 5.32e3T^{2} \)
79 \( 1 - 26.6iT - 6.24e3T^{2} \)
83 \( 1 - 89.6T + 6.88e3T^{2} \)
89 \( 1 + 139. iT - 7.92e3T^{2} \)
97 \( 1 - 41.3iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.91730853618613267287216443240, −11.20982121605002583543731903114, −10.49884803410462024153992987413, −9.375438252650864957395614308871, −8.646421374985725085722764897092, −6.92903825188063280440792999785, −5.27188480111973732934442480420, −4.26663587973579902561832770909, −2.30710202936270248810263749947, −1.57031048507432707525184303792, 1.95913339886760219415152555833, 4.63700722592053482566942505594, 5.53376818634543006671066909452, 6.33426030285774146373839130343, 7.81852315991813074124482249632, 8.279916082240336691489313435432, 9.563538440400875167504238524339, 10.69296029847815626688437426376, 11.87701101250328972798216401173, 13.56698926598903453231733720761

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