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Properties

Label 2-171-171.112-c1-0-1
Degree $2$
Conductor $171$
Sign $-0.950 - 0.311i$
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  + (−0.524 + 0.190i)2-s + (−0.312 + 1.70i)3-s + (−1.29 + 1.08i)4-s + (−0.228 + 1.29i)5-s + (−0.161 − 0.952i)6-s + (0.589 − 1.02i)7-s + (1.02 − 1.78i)8-s + (−2.80 − 1.06i)9-s + (−0.127 − 0.723i)10-s − 5.56·11-s + (−1.44 − 2.54i)12-s + (0.432 + 2.45i)13-s + (−0.114 + 0.647i)14-s + (−2.13 − 0.795i)15-s + (0.387 − 2.19i)16-s + (−0.793 + 4.50i)17-s + ⋯
L(s)  = 1  + (−0.370 + 0.134i)2-s + (−0.180 + 0.983i)3-s + (−0.646 + 0.542i)4-s + (−0.102 + 0.580i)5-s + (−0.0657 − 0.388i)6-s + (0.222 − 0.385i)7-s + (0.363 − 0.630i)8-s + (−0.934 − 0.355i)9-s + (−0.0403 − 0.228i)10-s − 1.67·11-s + (−0.417 − 0.734i)12-s + (0.119 + 0.679i)13-s + (−0.0305 + 0.173i)14-s + (−0.552 − 0.205i)15-s + (0.0967 − 0.548i)16-s + (−0.192 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.950 - 0.311i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.950 - 0.311i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0882562 + 0.552085i\)
\(L(\frac12)\) \(\approx\) \(0.0882562 + 0.552085i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.312 - 1.70i)T \)
19 \( 1 + (-3.50 - 2.59i)T \)
good2 \( 1 + (0.524 - 0.190i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (0.228 - 1.29i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.589 + 1.02i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 5.56T + 11T^{2} \)
13 \( 1 + (-0.432 - 2.45i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.793 - 4.50i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-1.00 + 0.845i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (5.09 - 4.27i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 - 3.66T + 31T^{2} \)
37 \( 1 + 8.31T + 37T^{2} \)
41 \( 1 + (0.310 - 0.112i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-6.15 - 5.16i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-5.17 + 4.34i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-2.82 - 1.02i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-8.01 - 6.72i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.136 + 0.776i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (14.3 + 5.21i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-1.11 + 0.406i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-3.80 - 3.19i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-1.60 + 9.10i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.814 + 1.41i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.78 + 2.34i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (12.5 - 4.56i)T + (74.3 - 62.3i)T^{2} \)
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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

−13.28322221780550999627830269256, −12.15355520723861090091181958964, −10.76783923221239673141268057171, −10.37372839616385710391104812902, −9.177251195550841041797712190117, −8.213784603513513541537612561834, −7.19824201869748318693257764970, −5.50770058237442754666355528747, −4.32731384074639568349851458966, −3.18291202324604154099194923755, 0.60726010005902353665781312533, 2.51611823456033575456882658997, 5.08002835217849638652972399383, 5.52856759607437825747753035489, 7.34638554407890366630699621959, 8.245078925987315323146701110091, 9.089495436759949033025262111844, 10.35995389823833537004234078474, 11.32022354044517816912205748844, 12.42745745816791735055940457055

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