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Properties

Label 2-171-171.121-c1-0-12
Degree $2$
Conductor $171$
Sign $-0.156 + 0.987i$
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  + (−1.04 + 1.81i)2-s + (0.154 − 1.72i)3-s + (−1.20 − 2.07i)4-s − 2.89·5-s + (2.97 + 2.08i)6-s + (0.116 + 0.201i)7-s + 0.839·8-s + (−2.95 − 0.532i)9-s + (3.03 − 5.26i)10-s + (−1.99 − 3.45i)11-s + (−3.77 + 1.74i)12-s + (−1.91 − 3.32i)13-s − 0.488·14-s + (−0.446 + 4.99i)15-s + (1.51 − 2.63i)16-s + (0.0780 + 0.135i)17-s + ⋯
L(s)  = 1  + (−0.741 + 1.28i)2-s + (0.0890 − 0.996i)3-s + (−0.600 − 1.03i)4-s − 1.29·5-s + (1.21 + 0.853i)6-s + (0.0440 + 0.0762i)7-s + 0.296·8-s + (−0.984 − 0.177i)9-s + (0.960 − 1.66i)10-s + (−0.601 − 1.04i)11-s + (−1.08 + 0.505i)12-s + (−0.531 − 0.921i)13-s − 0.130·14-s + (−0.115 + 1.28i)15-s + (0.379 − 0.658i)16-s + (0.0189 + 0.0328i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156513 - 0.183215i\)
\(L(\frac12)\) \(\approx\) \(0.156513 - 0.183215i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.154 + 1.72i)T \)
19 \( 1 + (1.94 - 3.89i)T \)
good2 \( 1 + (1.04 - 1.81i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 2.89T + 5T^{2} \)
7 \( 1 + (-0.116 - 0.201i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.99 + 3.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.91 + 3.32i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.0780 - 0.135i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.471 - 0.815i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.25T + 29T^{2} \)
31 \( 1 + (-2.40 + 4.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 - 0.107T + 41T^{2} \)
43 \( 1 + (5.47 - 9.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.79T + 47T^{2} \)
53 \( 1 + (-4.03 + 6.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 4.98T + 61T^{2} \)
67 \( 1 + (3.56 + 6.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.33 + 5.77i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.38 - 9.32i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.10 + 14.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.46 + 9.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.25 + 2.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.04 - 1.80i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.43337866956995648011068047320, −11.67477118315818935841482154880, −10.38206061805257614101151781615, −8.733767955087862659926734176744, −8.040567362240668287368839336074, −7.61004785696995516817749286217, −6.44357447919485533253761193970, −5.38196336431755796693762389594, −3.21421338344590184650627107790, −0.26880266395706177953600739531, 2.54090442922559051767145746107, 3.88843039064678606452179943840, 4.83025741265742285330735168667, 7.12616088487796065205165625578, 8.423193505041256842136349957461, 9.180986955870859144351695121096, 10.27131649295956154707175839154, 10.85123700889747953217136737285, 11.91183479963296979930005870005, 12.31033049674313830138218010994

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