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Properties

Label 2-171-19.8-c2-0-5
Degree $2$
Conductor $171$
Sign $-0.217 + 0.975i$
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  + (−3.08 − 1.78i)2-s + (4.34 + 7.51i)4-s + (−2.32 + 4.03i)5-s − 10.6·7-s − 16.6i·8-s + (14.3 − 8.28i)10-s + 6.37·11-s + (15.3 − 8.87i)13-s + (32.9 + 19.0i)14-s + (−12.3 + 21.3i)16-s + (5.84 − 10.1i)17-s + (−3.88 − 18.5i)19-s − 40.4·20-s + (−19.6 − 11.3i)22-s + (−13.9 − 24.2i)23-s + ⋯
L(s)  = 1  + (−1.54 − 0.890i)2-s + (1.08 + 1.87i)4-s + (−0.465 + 0.806i)5-s − 1.52·7-s − 2.08i·8-s + (1.43 − 0.828i)10-s + 0.579·11-s + (1.18 − 0.682i)13-s + (2.35 + 1.35i)14-s + (−0.770 + 1.33i)16-s + (0.344 − 0.596i)17-s + (−0.204 − 0.978i)19-s − 2.02·20-s + (−0.893 − 0.515i)22-s + (−0.607 − 1.05i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 + 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.217 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.217 + 0.975i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ -0.217 + 0.975i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.279730 - 0.349086i\)
\(L(\frac12)\) \(\approx\) \(0.279730 - 0.349086i\)
\(L(2)\) 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 \( 1 + (3.88 + 18.5i)T \)
good2 \( 1 + (3.08 + 1.78i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (2.32 - 4.03i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 10.6T + 49T^{2} \)
11 \( 1 - 6.37T + 121T^{2} \)
13 \( 1 + (-15.3 + 8.87i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-5.84 + 10.1i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (13.9 + 24.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-33.2 + 19.1i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 42.9iT - 961T^{2} \)
37 \( 1 + 33.9iT - 1.36e3T^{2} \)
41 \( 1 + (16.3 + 9.45i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-26.5 + 45.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-12.0 - 20.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-13.3 + 7.69i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (25.6 + 14.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (21.3 + 36.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (15.1 - 8.75i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-74.6 - 43.0i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-46.2 + 80.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (26.3 + 15.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 77.1T + 6.88e3T^{2} \)
89 \( 1 + (-76.8 + 44.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (1.82 + 1.05i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.97038340747528574928849817947, −10.87061335406567458612486101913, −10.33978542507121563006316723812, −9.303065541582765215057889731567, −8.487445878679808194718623650246, −7.18693728918184101662168182711, −6.40167156459776104393174339111, −3.58763258911429292213201991256, −2.77423202117855275000312300353, −0.53240916620361376618686666712, 1.19340417236677942572499899424, 3.85246548139193151892848726746, 5.98589215592205723275501559495, 6.55223247387206898285596717931, 7.903673800781625013406549141047, 8.722412056415745431175647282601, 9.520412941700267349068858204045, 10.30202773764148125688410939728, 11.64622378979675992567997531111, 12.71425953568062580007847002854

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