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Properties

Label 2-171-171.103-c2-0-11
Degree $2$
Conductor $171$
Sign $0.756 - 0.654i$
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  − 1.31i·2-s + (−0.588 + 2.94i)3-s + 2.28·4-s + (2.70 + 4.68i)5-s + (3.85 + 0.771i)6-s + (−2.07 − 3.59i)7-s − 8.23i·8-s + (−8.30 − 3.46i)9-s + (6.14 − 3.54i)10-s + (10.0 + 17.3i)11-s + (−1.34 + 6.71i)12-s + 17.6i·13-s + (−4.70 + 2.71i)14-s + (−15.3 + 5.20i)15-s − 1.66·16-s + (3.78 − 6.55i)17-s + ⋯
L(s)  = 1  − 0.655i·2-s + (−0.196 + 0.980i)3-s + 0.570·4-s + (0.541 + 0.937i)5-s + (0.642 + 0.128i)6-s + (−0.296 − 0.512i)7-s − 1.02i·8-s + (−0.923 − 0.384i)9-s + (0.614 − 0.354i)10-s + (0.909 + 1.57i)11-s + (−0.111 + 0.559i)12-s + 1.35i·13-s + (−0.336 + 0.194i)14-s + (−1.02 + 0.346i)15-s − 0.104·16-s + (0.222 − 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.756 - 0.654i$
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: \((2,\ 171,\ (\ :1),\ 0.756 - 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.60450 + 0.597605i\)
\(L(\frac12)\) \(\approx\) \(1.60450 + 0.597605i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.588 - 2.94i)T \)
19 \( 1 + (18.5 + 3.99i)T \)
good2 \( 1 + 1.31iT - 4T^{2} \)
5 \( 1 + (-2.70 - 4.68i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (2.07 + 3.59i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-10.0 - 17.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 17.6iT - 169T^{2} \)
17 \( 1 + (-3.78 + 6.55i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 31.7T + 529T^{2} \)
29 \( 1 + (-15.4 - 8.90i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (16.2 + 9.37i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 9.00iT - 1.36e3T^{2} \)
41 \( 1 + (37.5 - 21.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 48.2T + 1.84e3T^{2} \)
47 \( 1 + (-24.4 + 42.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-38.1 + 22.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-5.07 + 2.93i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-27.0 + 46.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + 118. iT - 4.48e3T^{2} \)
71 \( 1 + (88.8 + 51.2i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-34.0 + 59.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 62.5iT - 6.24e3T^{2} \)
83 \( 1 + (-35.2 - 60.9i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (97.7 - 56.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 113. iT - 9.40e3T^{2} \)
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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

−12.25983215323957733749267162688, −11.46561093402236249820462344409, −10.57821737907207742455484616584, −9.922592492983556035955727351718, −9.162854261498190066149336050990, −6.81095874677200725335723694918, −6.69817035394713739226273132439, −4.63511598142092377115600393262, −3.48505716268183568245654503415, −2.07361217120356734469897269684, 1.18360134706849199266802466771, 2.90774338017199235907794053296, 5.52474520566621459379011815665, 5.90115124143827091939582557052, 7.00010196517070116527499440146, 8.439470935357800072884884505626, 8.721907306700636266427341806728, 10.62679438703975003256177397032, 11.59155844035677420829134217390, 12.58555600152386945697324954520

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