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Properties

Label 2-171-171.103-c2-0-18
Degree $2$
Conductor $171$
Sign $0.147 - 0.989i$
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  + 0.410i·2-s + (1.62 + 2.52i)3-s + 3.83·4-s + (4.20 + 7.27i)5-s + (−1.03 + 0.666i)6-s + (−1.03 − 1.79i)7-s + 3.21i·8-s + (−3.72 + 8.19i)9-s + (−2.98 + 1.72i)10-s + (−9.88 − 17.1i)11-s + (6.22 + 9.66i)12-s − 19.5i·13-s + (0.735 − 0.424i)14-s + (−11.5 + 22.4i)15-s + 14.0·16-s + (−3.36 + 5.82i)17-s + ⋯
L(s)  = 1  + 0.205i·2-s + (0.541 + 0.840i)3-s + 0.957·4-s + (0.840 + 1.45i)5-s + (−0.172 + 0.111i)6-s + (−0.147 − 0.256i)7-s + 0.401i·8-s + (−0.413 + 0.910i)9-s + (−0.298 + 0.172i)10-s + (−0.898 − 1.55i)11-s + (0.518 + 0.805i)12-s − 1.50i·13-s + (0.0525 − 0.0303i)14-s + (−0.768 + 1.49i)15-s + 0.875·16-s + (−0.197 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.147 - 0.989i$
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: \((2,\ 171,\ (\ :1),\ 0.147 - 0.989i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.69743 + 1.46266i\)
\(L(\frac12)\) \(\approx\) \(1.69743 + 1.46266i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 - 2.52i)T \)
19 \( 1 + (9.59 + 16.3i)T \)
good2 \( 1 - 0.410iT - 4T^{2} \)
5 \( 1 + (-4.20 - 7.27i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (1.03 + 1.79i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (9.88 + 17.1i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 19.5iT - 169T^{2} \)
17 \( 1 + (3.36 - 5.82i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 9.49T + 529T^{2} \)
29 \( 1 + (-37.1 - 21.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (3.02 + 1.74i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 5.01iT - 1.36e3T^{2} \)
41 \( 1 + (-42.9 + 24.8i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 26.2T + 1.84e3T^{2} \)
47 \( 1 + (33.7 - 58.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-0.436 + 0.252i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (21.4 - 12.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (32.6 - 56.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + 36.9iT - 4.48e3T^{2} \)
71 \( 1 + (52.8 + 30.5i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (30.2 - 52.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 9.81iT - 6.24e3T^{2} \)
83 \( 1 + (9.60 + 16.6i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (87.8 - 50.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 95.7iT - 9.40e3T^{2} \)
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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

−13.07737124613339627938609701281, −11.11188346149580544450242404215, −10.67783070230850781782842071160, −10.19703534277505419043404170114, −8.590000745865207472374200593768, −7.55460205466191875040781035007, −6.31920813463756264234534684881, −5.45372542508337047473744818011, −3.10423502411983608092515921837, −2.74789250583428676238800656519, 1.59890413212796867965482278170, 2.36551475236895211322692639213, 4.58577639369931629323941312318, 6.04778874071996701764224247164, 7.04267403153691811203464199049, 8.186325371724003434623633865625, 9.287558317164389776172420616990, 10.05091665171842187661259780815, 11.74837578244621880316652478626, 12.44804810810378227671715480392

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