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Properties

Label 2-171-171.106-c1-0-6
Degree $2$
Conductor $171$
Sign $-0.823 - 0.567i$
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  + (1.19 + 2.07i)2-s + (−0.340 + 1.69i)3-s + (−1.87 + 3.25i)4-s + 0.719·5-s + (−3.93 + 1.32i)6-s + (1.65 − 2.87i)7-s − 4.21·8-s + (−2.76 − 1.15i)9-s + (0.863 + 1.49i)10-s + (0.550 − 0.953i)11-s + (−4.88 − 4.29i)12-s + (2.37 − 4.11i)13-s + 7.95·14-s + (−0.245 + 1.22i)15-s + (−1.30 − 2.25i)16-s + (−3.13 + 5.42i)17-s + ⋯
L(s)  = 1  + (0.848 + 1.46i)2-s + (−0.196 + 0.980i)3-s + (−0.939 + 1.62i)4-s + 0.321·5-s + (−1.60 + 0.542i)6-s + (0.626 − 1.08i)7-s − 1.49·8-s + (−0.922 − 0.385i)9-s + (0.273 + 0.472i)10-s + (0.165 − 0.287i)11-s + (−1.41 − 1.24i)12-s + (0.658 − 1.14i)13-s + 2.12·14-s + (−0.0633 + 0.315i)15-s + (−0.325 − 0.563i)16-s + (−0.760 + 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.823 - 0.567i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.493988 + 1.58755i\)
\(L(\frac12)\) \(\approx\) \(0.493988 + 1.58755i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.340 - 1.69i)T \)
19 \( 1 + (4.19 - 1.19i)T \)
good2 \( 1 + (-1.19 - 2.07i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 0.719T + 5T^{2} \)
7 \( 1 + (-1.65 + 2.87i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.550 + 0.953i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.37 + 4.11i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.13 - 5.42i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.11 + 1.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.94T + 29T^{2} \)
31 \( 1 + (0.763 + 1.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.69T + 37T^{2} \)
41 \( 1 - 5.68T + 41T^{2} \)
43 \( 1 + (2.30 + 3.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.283T + 47T^{2} \)
53 \( 1 + (-1.90 - 3.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 + (-3.72 + 6.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.51 + 9.55i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.22 - 9.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.11 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.05 - 8.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.23 - 7.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.16 + 10.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.57508907447461941974699344498, −12.58807519592975222514729699539, −10.92513077816696670205367054975, −10.36464659207737544057531203958, −8.650173607479209864031948668912, −7.989439754298503921707991415022, −6.49243185657893376974882930777, −5.71339001121300950869160121709, −4.49475710327114116470335885855, −3.74549619462797500006416317594, 1.72601202467479696482538331985, 2.60199320321228723014279149648, 4.50710414176523956039562833867, 5.59461699206206810997938308988, 6.79201210215860231563627590486, 8.550498871733603922199897906294, 9.470015730216847030900843176090, 11.01185213075376584153326656140, 11.60019904322486000140547713766, 12.18917088804418507056246076218

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