Properties

Label 2-171-171.106-c1-0-15
Degree $2$
Conductor $171$
Sign $0.322 + 0.946i$
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  + (−0.0732 − 0.126i)2-s + (0.157 − 1.72i)3-s + (0.989 − 1.71i)4-s + 2.57·5-s + (−0.230 + 0.106i)6-s + (−1.73 + 3.01i)7-s − 0.582·8-s + (−2.95 − 0.543i)9-s + (−0.188 − 0.326i)10-s + (2.07 − 3.60i)11-s + (−2.79 − 1.97i)12-s + (−2.29 + 3.98i)13-s + 0.509·14-s + (0.404 − 4.43i)15-s + (−1.93 − 3.35i)16-s + (−1.50 + 2.61i)17-s + ⋯
L(s)  = 1  + (−0.0518 − 0.0897i)2-s + (0.0908 − 0.995i)3-s + (0.494 − 0.856i)4-s + 1.14·5-s + (−0.0940 + 0.0434i)6-s + (−0.657 + 1.13i)7-s − 0.206·8-s + (−0.983 − 0.181i)9-s + (−0.0595 − 0.103i)10-s + (0.626 − 1.08i)11-s + (−0.808 − 0.570i)12-s + (−0.637 + 1.10i)13-s + 0.136·14-s + (0.104 − 1.14i)15-s + (−0.483 − 0.838i)16-s + (−0.365 + 0.633i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.322 + 0.946i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.322 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10144 - 0.788632i\)
\(L(\frac12)\) \(\approx\) \(1.10144 - 0.788632i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.157 + 1.72i)T \)
19 \( 1 + (-3.65 - 2.37i)T \)
good2 \( 1 + (0.0732 + 0.126i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 2.57T + 5T^{2} \)
7 \( 1 + (1.73 - 3.01i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.07 + 3.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.29 - 3.98i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.50 - 2.61i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.45 + 4.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.71T + 29T^{2} \)
31 \( 1 + (-3.31 - 5.75i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.28T + 37T^{2} \)
41 \( 1 - 2.53T + 41T^{2} \)
43 \( 1 + (2.39 + 4.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.71T + 47T^{2} \)
53 \( 1 + (-5.35 - 9.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.96T + 59T^{2} \)
61 \( 1 - 0.944T + 61T^{2} \)
67 \( 1 + (0.688 - 1.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.45 + 7.71i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.14 - 3.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.69 + 2.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.52 + 7.83i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.96 + 6.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.930 + 1.61i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.42002070611476717783028074970, −11.78830067774959896604581969329, −10.59521524564033128234788394349, −9.322098454359142896668991040758, −8.839114393447591674291120528324, −6.92231462738879812049578221862, −6.14042051413142075949608931080, −5.56606744009764474886291569487, −2.75610333043529341200519466484, −1.66155640910534132401050105198, 2.67047100413379607051051434545, 3.91605895325183083578901973633, 5.31772834375181186786462443439, 6.77050694628980325777263766757, 7.65597655942940894256597168302, 9.357555265876191776636721665737, 9.769231715995523333626004355430, 10.77547905961646670563530712057, 11.90608257964112049335563586250, 13.14473163019632781527981615756

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