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Properties

Label 2-171-19.11-c1-0-0
Degree $2$
Conductor $171$
Sign $-0.567 + 0.823i$
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  + (−1.16 + 2.02i)2-s + (−1.72 − 2.98i)4-s + (−0.524 + 0.908i)5-s − 3.44·7-s + 3.38·8-s + (−1.22 − 2.12i)10-s − 5.71·11-s + (0.5 + 0.866i)13-s + (4.02 − 6.97i)14-s + (−0.500 + 0.866i)16-s + (1.04 − 1.81i)17-s + (1 + 4.24i)19-s + 3.61·20-s + (6.67 − 11.5i)22-s + (1.80 + 3.13i)23-s + ⋯
L(s)  = 1  + (−0.825 + 1.42i)2-s + (−0.862 − 1.49i)4-s + (−0.234 + 0.406i)5-s − 1.30·7-s + 1.19·8-s + (−0.387 − 0.670i)10-s − 1.72·11-s + (0.138 + 0.240i)13-s + (1.07 − 1.86i)14-s + (−0.125 + 0.216i)16-s + (0.254 − 0.440i)17-s + (0.229 + 0.973i)19-s + 0.809·20-s + (1.42 − 2.46i)22-s + (0.377 + 0.653i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.567 + 0.823i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.567 + 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105361 - 0.200437i\)
\(L(\frac12)\) \(\approx\) \(0.105361 - 0.200437i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 4.24i)T \)
good2 \( 1 + (1.16 - 2.02i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.524 - 0.908i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 + 5.71T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.04 + 1.81i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.80 - 3.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.61 + 6.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.44T + 31T^{2} \)
37 \( 1 - 3.89T + 37T^{2} \)
41 \( 1 + (4.66 - 8.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.17 - 5.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.66 - 8.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.524 - 0.908i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.90 + 6.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.174 + 0.301i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.61 + 6.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.174 - 0.301i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + (2.62 + 4.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.55 - 2.68i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.52469750368033338418998966224, −12.72262785535506406658835968842, −11.11468805809149319121343795866, −9.927738319988938078742900261705, −9.374323007642214723034162031078, −7.945723517438357304902467071890, −7.36556683936516710509513459177, −6.23916253747487226213303530638, −5.32359514577515486715068028075, −3.20446852748645676549388394871, 0.25852575978823346909573396353, 2.53012790942622937604874157457, 3.59951872493571199716439445806, 5.37991315035471063801366666066, 7.16760463541432259765514975770, 8.475256643206997600284867986081, 9.224340233120072038634157303472, 10.37465434400901984673741307918, 10.74780540115663247756279378579, 12.14330004486223150510423024135

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