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Properties

Label 2-171-171.14-c1-0-14
Degree $2$
Conductor $171$
Sign $0.926 + 0.376i$
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  + (0.469 + 0.393i)2-s + (1.72 + 0.105i)3-s + (−0.282 − 1.59i)4-s + (−0.821 − 2.25i)5-s + (0.770 + 0.730i)6-s + (−0.676 + 1.17i)7-s + (1.11 − 1.92i)8-s + (2.97 + 0.363i)9-s + (0.503 − 1.38i)10-s − 0.212i·11-s + (−0.319 − 2.79i)12-s + (−2.01 + 5.54i)13-s + (−0.779 + 0.283i)14-s + (−1.18 − 3.98i)15-s + (−1.77 + 0.645i)16-s + (0.897 + 2.46i)17-s + ⋯
L(s)  = 1  + (0.331 + 0.278i)2-s + (0.998 + 0.0607i)3-s + (−0.141 − 0.799i)4-s + (−0.367 − 1.00i)5-s + (0.314 + 0.298i)6-s + (−0.255 + 0.443i)7-s + (0.392 − 0.680i)8-s + (0.992 + 0.121i)9-s + (0.159 − 0.437i)10-s − 0.0642i·11-s + (−0.0921 − 0.806i)12-s + (−0.559 + 1.53i)13-s + (−0.208 + 0.0758i)14-s + (−0.305 − 1.02i)15-s + (−0.443 + 0.161i)16-s + (0.217 + 0.598i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.926 + 0.376i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.926 + 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60329 - 0.313118i\)
\(L(\frac12)\) \(\approx\) \(1.60329 - 0.313118i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.105i)T \)
19 \( 1 + (0.154 - 4.35i)T \)
good2 \( 1 + (-0.469 - 0.393i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (0.821 + 2.25i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (0.676 - 1.17i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.212iT - 11T^{2} \)
13 \( 1 + (2.01 - 5.54i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.897 - 2.46i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-1.61 + 0.284i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (0.914 + 5.18i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 - 3.57iT - 31T^{2} \)
37 \( 1 + 8.01iT - 37T^{2} \)
41 \( 1 + (6.91 + 5.80i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.233 - 1.32i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.80 + 0.494i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (7.65 - 6.42i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-1.30 + 7.39i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (4.70 + 1.71i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (6.37 + 7.59i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (7.93 + 6.65i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.33 - 7.55i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.92 + 5.28i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-14.1 - 8.15i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.680 + 3.85i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-4.16 + 4.96i)T + (-16.8 - 95.5i)T^{2} \)
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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

−12.79783710955610595656189757382, −12.07728634282316802628861130734, −10.46294363602221635370328588649, −9.370992812554062303352349256352, −8.851783207167012676018897161339, −7.57957273014072201211592941646, −6.27126377103259642985727005249, −4.86954151176685861793536300144, −3.94743263083450758237775657296, −1.79696443574428153813439414368, 2.84300418888860360164316331863, 3.32217081473803442717073942323, 4.79406247655445390555932565871, 7.00865289047948613452355219220, 7.55281303876910605249969445944, 8.583918032314909232762830448873, 9.896110658277904936994356350392, 10.84227955323533270512304864597, 11.97514135303576606313477926290, 13.08997490840125828533752643016

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