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Properties

Label 2-171-171.106-c1-0-4
Degree $2$
Conductor $171$
Sign $-0.0710 - 0.997i$
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  + (0.616 + 1.06i)2-s + (0.506 + 1.65i)3-s + (0.239 − 0.414i)4-s − 0.551·5-s + (−1.45 + 1.56i)6-s + (−1.62 + 2.80i)7-s + 3.05·8-s + (−2.48 + 1.67i)9-s + (−0.340 − 0.589i)10-s + (2.68 − 4.65i)11-s + (0.807 + 0.186i)12-s + (−1.76 + 3.05i)13-s − 4.00·14-s + (−0.279 − 0.913i)15-s + (1.40 + 2.43i)16-s + (2.60 − 4.50i)17-s + ⋯
L(s)  = 1  + (0.436 + 0.755i)2-s + (0.292 + 0.956i)3-s + (0.119 − 0.207i)4-s − 0.246·5-s + (−0.594 + 0.638i)6-s + (−0.612 + 1.06i)7-s + 1.08·8-s + (−0.828 + 0.559i)9-s + (−0.107 − 0.186i)10-s + (0.810 − 1.40i)11-s + (0.233 + 0.0537i)12-s + (−0.488 + 0.846i)13-s − 1.06·14-s + (−0.0721 − 0.235i)15-s + (0.351 + 0.609i)16-s + (0.630 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0710 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0710 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.0710 - 0.997i$
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: \((2,\ 171,\ (\ :1/2),\ -0.0710 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05242 + 1.13007i\)
\(L(\frac12)\) \(\approx\) \(1.05242 + 1.13007i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.506 - 1.65i)T \)
19 \( 1 + (-0.164 + 4.35i)T \)
good2 \( 1 + (-0.616 - 1.06i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 0.551T + 5T^{2} \)
7 \( 1 + (1.62 - 2.80i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.68 + 4.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.76 - 3.05i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.60 + 4.50i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.49 - 2.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.49T + 29T^{2} \)
31 \( 1 + (2.54 + 4.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.20T + 37T^{2} \)
41 \( 1 - 1.71T + 41T^{2} \)
43 \( 1 + (-1.79 - 3.11i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + (-0.562 - 0.973i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.77T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + (1.18 - 2.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.507 - 0.879i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.98 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.568 - 0.985i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.14 - 1.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.12 - 1.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.09 + 7.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.48817528300189967305754751438, −11.75396588264101507146311485762, −11.23691465404807621426709486994, −9.689706618199057540441174594374, −9.147068136725546420553784264214, −7.83838420006591589601089982239, −6.37342798655032689293035936421, −5.57198697120794280564783403354, −4.36733589208742647816949273650, −2.87524875358516500282006768066, 1.63745382107262464065711448396, 3.25588635941600057399310177910, 4.27058666586534429632172274832, 6.33185948553077050173091033153, 7.42287944830095902003089338612, 7.999449886720337560695518616509, 9.808835218313912829675082849893, 10.58340457937072656538549331040, 12.03986769262935055597307363140, 12.41463637907200579797361301283

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