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Properties

Label 2-171-171.103-c2-0-1
Degree $2$
Conductor $171$
Sign $-0.892 - 0.451i$
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  + 2.41i·2-s + (−2.25 − 1.97i)3-s − 1.82·4-s + (3.12 + 5.41i)5-s + (4.77 − 5.43i)6-s + (−3.26 − 5.65i)7-s + 5.25i·8-s + (1.16 + 8.92i)9-s + (−13.0 + 7.53i)10-s + (1.18 + 2.04i)11-s + (4.11 + 3.61i)12-s + 17.7i·13-s + (13.6 − 7.87i)14-s + (3.67 − 18.3i)15-s − 19.9·16-s + (−6.18 + 10.7i)17-s + ⋯
L(s)  = 1  + 1.20i·2-s + (−0.751 − 0.659i)3-s − 0.455·4-s + (0.624 + 1.08i)5-s + (0.796 − 0.906i)6-s + (−0.466 − 0.807i)7-s + 0.656i·8-s + (0.128 + 0.991i)9-s + (−1.30 + 0.753i)10-s + (0.107 + 0.186i)11-s + (0.342 + 0.300i)12-s + 1.36i·13-s + (0.974 − 0.562i)14-s + (0.244 − 1.22i)15-s − 1.24·16-s + (−0.363 + 0.629i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.892 - 0.451i$
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: \((2,\ 171,\ (\ :1),\ -0.892 - 0.451i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.248536 + 1.04171i\)
\(L(\frac12)\) \(\approx\) \(0.248536 + 1.04171i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.25 + 1.97i)T \)
19 \( 1 + (17.8 + 6.45i)T \)
good2 \( 1 - 2.41iT - 4T^{2} \)
5 \( 1 + (-3.12 - 5.41i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (3.26 + 5.65i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-1.18 - 2.04i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 17.7iT - 169T^{2} \)
17 \( 1 + (6.18 - 10.7i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 18.2T + 529T^{2} \)
29 \( 1 + (-20.6 - 11.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-36.0 - 20.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 17.7iT - 1.36e3T^{2} \)
41 \( 1 + (-35.4 + 20.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 0.656T + 1.84e3T^{2} \)
47 \( 1 + (21.7 - 37.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-56.0 + 32.3i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-78.7 + 45.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (19.7 - 34.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 114. iT - 4.48e3T^{2} \)
71 \( 1 + (7.60 + 4.39i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-68.3 + 118. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 131. iT - 6.24e3T^{2} \)
83 \( 1 + (12.4 + 21.4i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (11.2 - 6.52i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 118. iT - 9.40e3T^{2} \)
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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

−13.30622524315436778754189405299, −11.92234626007814423903105299370, −10.89275385332401495828482614716, −10.12735421440830628150893054085, −8.504305536684918698398274739036, −7.16659187906010397032040163209, −6.62731008467218335279071465191, −6.13108177494587971003364799354, −4.50385542965970666370355615238, −2.18171935776134123963083789695, 0.70114233940892286380693625014, 2.59468005210923838779707038612, 4.15888720736773721422752041176, 5.42454936856161875650685823921, 6.32782686218354554803502718137, 8.505371342126606773520679561970, 9.548107569475654332481170616350, 10.10198521099303049483319056435, 11.13694544258016754753407037297, 12.19308327548199371288411789572

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