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Properties

Label 2-171-171.103-c2-0-0
Degree $2$
Conductor $171$
Sign $0.593 - 0.804i$
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  − 3.71i·2-s + (−1.39 + 2.65i)3-s − 9.80·4-s + (−4.33 − 7.50i)5-s + (9.87 + 5.17i)6-s + (3.03 + 5.26i)7-s + 21.5i·8-s + (−5.12 − 7.39i)9-s + (−27.8 + 16.0i)10-s + (3.10 + 5.38i)11-s + (13.6 − 26.0i)12-s + 14.0i·13-s + (19.5 − 11.2i)14-s + (25.9 − 1.06i)15-s + 40.9·16-s + (−1.83 + 3.17i)17-s + ⋯
L(s)  = 1  − 1.85i·2-s + (−0.463 + 0.885i)3-s − 2.45·4-s + (−0.866 − 1.50i)5-s + (1.64 + 0.861i)6-s + (0.434 + 0.751i)7-s + 2.69i·8-s + (−0.569 − 0.821i)9-s + (−2.78 + 1.60i)10-s + (0.282 + 0.489i)11-s + (1.13 − 2.17i)12-s + 1.08i·13-s + (1.39 − 0.806i)14-s + (1.73 − 0.0712i)15-s + 2.55·16-s + (−0.107 + 0.186i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.593 - 0.804i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ 0.593 - 0.804i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.120203 + 0.0607054i\)
\(L(\frac12)\) \(\approx\) \(0.120203 + 0.0607054i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.39 - 2.65i)T \)
19 \( 1 + (15.8 - 10.4i)T \)
good2 \( 1 + 3.71iT - 4T^{2} \)
5 \( 1 + (4.33 + 7.50i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.03 - 5.26i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.10 - 5.38i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 14.0iT - 169T^{2} \)
17 \( 1 + (1.83 - 3.17i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 5.63T + 529T^{2} \)
29 \( 1 + (44.1 + 25.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (39.6 + 22.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 9.96iT - 1.36e3T^{2} \)
41 \( 1 + (34.7 - 20.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 20.9T + 1.84e3T^{2} \)
47 \( 1 + (24.7 - 42.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (50.5 - 29.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-26.1 + 15.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (3.80 - 6.58i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + 49.1iT - 4.48e3T^{2} \)
71 \( 1 + (-68.0 - 39.3i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (16.3 - 28.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 155. iT - 6.24e3T^{2} \)
83 \( 1 + (59.1 + 102. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (21.5 - 12.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 19.6iT - 9.40e3T^{2} \)
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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

−12.35934746974584681813604662933, −11.56101797184349970791174233986, −11.13025036935192462498787913995, −9.598164521156761801283017271216, −9.133527932793189594764588031457, −8.270628415014251992014933942126, −5.47190321159137977217341910254, −4.43339454231665468901806411368, −3.91828227898925351621357402522, −1.77619184114559741274533524215, 0.088657462870037458426185347006, 3.59150684751323264338340899350, 5.20174781239384197423823690316, 6.40830540755912358287062631128, 7.17695500341624198146761530041, 7.64635233625155643356783042525, 8.644395381090362447775033592730, 10.55803268965474062235502591064, 11.23564091182840486781492271712, 12.78198458788098823229208958169

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