Properties

Label 2-171-19.12-c2-0-8
Degree $2$
Conductor $171$
Sign $0.338 + 0.940i$
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  + (−1.99 + 1.14i)2-s + (0.640 − 1.10i)4-s + (0.140 + 0.243i)5-s − 5.24·7-s − 6.24i·8-s + (−0.558 − 0.322i)10-s + 1.15·11-s + (−6.32 − 3.65i)13-s + (10.4 − 6.02i)14-s + (9.74 + 16.8i)16-s + (−7.52 − 13.0i)17-s + (1.52 − 18.9i)19-s + 0.359·20-s + (−2.30 + 1.32i)22-s + (13.3 − 23.1i)23-s + ⋯
L(s)  = 1  + (−0.995 + 0.574i)2-s + (0.160 − 0.277i)4-s + (0.0280 + 0.0486i)5-s − 0.748·7-s − 0.781i·8-s + (−0.0558 − 0.0322i)10-s + 0.105·11-s + (−0.486 − 0.280i)13-s + (0.745 − 0.430i)14-s + (0.608 + 1.05i)16-s + (−0.442 − 0.766i)17-s + (0.0801 − 0.996i)19-s + 0.0179·20-s + (−0.104 + 0.0604i)22-s + (0.581 − 1.00i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.338 + 0.940i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ 0.338 + 0.940i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.355385 - 0.249789i\)
\(L(\frac12)\) \(\approx\) \(0.355385 - 0.249789i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (-1.52 + 18.9i)T \)
good2 \( 1 + (1.99 - 1.14i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-0.140 - 0.243i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 5.24T + 49T^{2} \)
11 \( 1 - 1.15T + 121T^{2} \)
13 \( 1 + (6.32 + 3.65i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (7.52 + 13.0i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-13.3 + 23.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-7.76 - 4.48i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 11.7iT - 961T^{2} \)
37 \( 1 + 36.1iT - 1.36e3T^{2} \)
41 \( 1 + (40.3 - 23.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-1.64 - 2.84i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (26.3 - 45.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (88.0 + 50.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-39.8 + 22.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (50.7 - 87.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (50.9 + 29.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-90.7 + 52.3i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-63.1 - 109. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (55.9 - 32.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 22.0T + 6.88e3T^{2} \)
89 \( 1 + (107. + 62.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-162. + 93.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.48604514729001346255055800482, −11.09232908583042051294564528856, −9.962549424525084659648047355857, −9.242806013623952214345788438984, −8.305964954437610191710009227908, −7.10262706441250010810827399817, −6.43212382070534489431598265091, −4.67333057434990471743608274193, −2.94174367412872288221104885647, −0.36986254549519978206089716352, 1.62612383278772591596437753402, 3.31807540632195250449342338073, 5.13355208499972933254737860788, 6.51828991937071483147868387635, 7.84798964377039225085680674064, 8.940039797321472243615854969932, 9.715196197719328754295403823573, 10.51609081957318441752515149050, 11.52653171695119376463640292794, 12.49455378629159046925343471995

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