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Properties

Label 2-171-171.121-c1-0-3
Degree $2$
Conductor $171$
Sign $-0.758 - 0.652i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.803 + 1.39i)2-s + (−1.24 + 1.20i)3-s + (−0.290 − 0.503i)4-s + 3.75·5-s + (−0.686 − 2.69i)6-s + (2.27 + 3.94i)7-s − 2.27·8-s + (0.0758 − 2.99i)9-s + (−3.01 + 5.22i)10-s + (−1.29 − 2.24i)11-s + (0.969 + 0.272i)12-s + (0.268 + 0.465i)13-s − 7.32·14-s + (−4.66 + 4.54i)15-s + (2.41 − 4.17i)16-s + (−1.73 − 2.99i)17-s + ⋯
L(s)  = 1  + (−0.568 + 0.983i)2-s + (−0.715 + 0.698i)3-s + (−0.145 − 0.251i)4-s + 1.68·5-s + (−0.280 − 1.10i)6-s + (0.861 + 1.49i)7-s − 0.805·8-s + (0.0252 − 0.999i)9-s + (−0.954 + 1.65i)10-s + (−0.390 − 0.676i)11-s + (0.279 + 0.0787i)12-s + (0.0745 + 0.129i)13-s − 1.95·14-s + (−1.20 + 1.17i)15-s + (0.603 − 1.04i)16-s + (−0.419 − 0.726i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.758 - 0.652i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.758 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.320990 + 0.865300i\)
\(L(\frac12)\) \(\approx\) \(0.320990 + 0.865300i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.24 - 1.20i)T \)
19 \( 1 + (4.01 - 1.69i)T \)
good2 \( 1 + (0.803 - 1.39i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 3.75T + 5T^{2} \)
7 \( 1 + (-2.27 - 3.94i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.29 + 2.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.268 - 0.465i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.73 + 2.99i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.104 - 0.180i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.853T + 29T^{2} \)
31 \( 1 + (-3.83 + 6.64i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.41T + 37T^{2} \)
41 \( 1 - 0.939T + 41T^{2} \)
43 \( 1 + (1.99 - 3.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.14T + 47T^{2} \)
53 \( 1 + (-5.68 + 9.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.40T + 59T^{2} \)
61 \( 1 + 7.18T + 61T^{2} \)
67 \( 1 + (0.140 + 0.242i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.43 - 5.94i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.416 + 0.721i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.91 + 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.63 - 6.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.20 - 5.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.18 + 5.50i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.19689520724758312874662274050, −11.98198413914943345016403607306, −11.08472043708040397778396978002, −9.781745378820616047428728230819, −9.096002116251252723813088661237, −8.267518634852191076530983412007, −6.41725921482370657872237111983, −5.85300335859604327357563646372, −5.07204937370007277903900117522, −2.48854823433775371261169309234, 1.28547465497945700912513999533, 2.21892353384872660266394271842, 4.73105101942256371983098097608, 6.04440792893896337705444106977, 7.01028005542026168589696656553, 8.440739265512573569185786801463, 9.849499552931275437412889301075, 10.63425724810781036701953887038, 10.89483487835688161299416828134, 12.33819360928379798739547494339

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