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Properties

Label 2-171-19.6-c1-0-6
Degree $2$
Conductor $171$
Sign $0.173 + 0.984i$
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  + (1.93 − 1.62i)2-s + (0.766 − 4.34i)4-s + (0.233 + 1.32i)5-s + (−0.766 + 1.32i)7-s + (−3.05 − 5.28i)8-s + (2.61 + 2.19i)10-s + (−0.592 − 1.02i)11-s + (−2.55 + 0.929i)13-s + (0.673 + 3.82i)14-s + (−6.23 − 2.27i)16-s + (−2.97 + 2.49i)17-s + (0.819 + 4.28i)19-s + 5.94·20-s + (−2.81 − 1.02i)22-s + (0.879 − 4.98i)23-s + ⋯
L(s)  = 1  + (1.37 − 1.15i)2-s + (0.383 − 2.17i)4-s + (0.104 + 0.593i)5-s + (−0.289 + 0.501i)7-s + (−1.07 − 1.86i)8-s + (0.826 + 0.693i)10-s + (−0.178 − 0.309i)11-s + (−0.708 + 0.257i)13-s + (0.180 + 1.02i)14-s + (−1.55 − 0.567i)16-s + (−0.720 + 0.604i)17-s + (0.187 + 0.982i)19-s + 1.32·20-s + (−0.601 − 0.218i)22-s + (0.183 − 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63121 - 1.36966i\)
\(L(\frac12)\) \(\approx\) \(1.63121 - 1.36966i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (-0.819 - 4.28i)T \)
good2 \( 1 + (-1.93 + 1.62i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (-0.233 - 1.32i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.766 - 1.32i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.592 + 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.55 - 0.929i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.97 - 2.49i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.879 + 4.98i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-3.56 - 2.99i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.91 + 3.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.10T + 37T^{2} \)
41 \( 1 + (9.38 + 3.41i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.51 + 8.57i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.439 + 0.368i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (0.511 - 2.89i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-3.01 + 2.52i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.784 - 4.44i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.97 + 2.49i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.20 - 6.83i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (5.75 + 2.09i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (9.21 + 3.35i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-6.15 + 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.27 - 0.829i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-5.64 + 4.73i)T + (16.8 - 95.5i)T^{2} \)
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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

−12.42008614026903412653702642934, −11.86304344790247852179172944464, −10.64520441794701408448074141550, −10.17897453201477424232573124864, −8.702073793735460202408165259791, −6.79649955064119837599752401641, −5.80353651430066844340108181757, −4.60551372072034493105906290170, −3.27988165925398567621723719548, −2.19116532688291954056194394923, 3.07113452957801522001180601965, 4.60050491860993522575368535099, 5.19858195969331748087401737125, 6.65920845300636116595169622116, 7.34194753829268557699729201144, 8.556436514947793029078446879724, 9.852012214604919587469054077035, 11.45961492625921395564490736697, 12.43731172821532571171973931197, 13.30371845642775159016249221537

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