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Properties

Label 2-171-19.4-c1-0-6
Degree $2$
Conductor $171$
Sign $0.891 + 0.452i$
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  + (0.826 + 0.300i)2-s + (−0.939 − 0.788i)4-s + (1.93 − 1.62i)5-s + (0.939 − 1.62i)7-s + (−1.41 − 2.45i)8-s + (2.09 − 0.761i)10-s + (1.70 + 2.95i)11-s + (−0.918 + 5.21i)13-s + (1.26 − 1.06i)14-s + (−0.00727 − 0.0412i)16-s + (1.55 + 0.565i)17-s + (−2.52 − 3.55i)19-s − 3.10·20-s + (0.520 + 2.95i)22-s + (−1.34 − 1.13i)23-s + ⋯
L(s)  = 1  + (0.584 + 0.212i)2-s + (−0.469 − 0.394i)4-s + (0.867 − 0.727i)5-s + (0.355 − 0.615i)7-s + (−0.501 − 0.868i)8-s + (0.661 − 0.240i)10-s + (0.514 + 0.890i)11-s + (−0.254 + 1.44i)13-s + (0.338 − 0.283i)14-s + (−0.00181 − 0.0103i)16-s + (0.376 + 0.137i)17-s + (−0.578 − 0.815i)19-s − 0.694·20-s + (0.111 + 0.629i)22-s + (−0.280 − 0.235i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.891 + 0.452i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.891 + 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48304 - 0.354662i\)
\(L(\frac12)\) \(\approx\) \(1.48304 - 0.354662i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.52 + 3.55i)T \)
good2 \( 1 + (-0.826 - 0.300i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (-1.93 + 1.62i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.939 + 1.62i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.70 - 2.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.918 - 5.21i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.55 - 0.565i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (1.34 + 1.13i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.25 - 1.18i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.971 - 1.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.837T + 37T^{2} \)
41 \( 1 + (-0.779 - 4.42i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-3.67 + 3.08i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-0.673 + 0.245i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-4.67 - 3.92i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (10.1 + 3.67i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-3.36 - 2.82i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (13.3 - 4.86i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-10.5 + 8.84i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.30 + 7.40i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.20 + 6.85i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.25 + 2.17i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.396 + 2.24i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-1.71 - 0.623i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.90719216570547148609504289495, −12.00322715633376500853118459618, −10.56914615963263356542997902095, −9.462255463994468446554827784173, −8.999241780430030046850782867622, −7.20164412371690028807550505476, −6.14413339385744743132458567892, −4.87117473576331072338934622647, −4.20080499553241161223733084564, −1.63717876856064597652706896129, 2.50830551622319798995888264377, 3.68373398267317245113656309106, 5.41339575509880581001719551117, 6.02799980109928115866806292457, 7.79633422132649317482898172823, 8.745782070812734221356392180677, 9.909562211726163327828490604770, 10.95792344364350234726628897861, 12.02508456137887696353455714037, 12.87844732319606083340975406470

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