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Properties

Label 2-171-171.49-c1-0-9
Degree $2$
Conductor $171$
Sign $-0.248 + 0.968i$
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  − 2-s + 1.73i·3-s − 4-s + (−1.5 − 2.59i)5-s − 1.73i·6-s + (−0.5 − 0.866i)7-s + 3·8-s − 2.99·9-s + (1.5 + 2.59i)10-s + (−2.5 − 4.33i)11-s − 1.73i·12-s + 2·13-s + (0.5 + 0.866i)14-s + (4.5 − 2.59i)15-s − 16-s + (2.5 − 4.33i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.999i·3-s − 0.5·4-s + (−0.670 − 1.16i)5-s − 0.707i·6-s + (−0.188 − 0.327i)7-s + 1.06·8-s − 0.999·9-s + (0.474 + 0.821i)10-s + (−0.753 − 1.30i)11-s − 0.499i·12-s + 0.554·13-s + (0.133 + 0.231i)14-s + (1.16 − 0.670i)15-s − 0.250·16-s + (0.606 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.248 + 0.968i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.248 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.207956 - 0.268053i\)
\(L(\frac12)\) \(\approx\) \(0.207956 - 0.268053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
19 \( 1 + (4 - 1.73i)T \)
good2 \( 1 + T + 2T^{2} \)
5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (1.5 - 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−12.39722657023686509638962492063, −11.18435209710687841091396181229, −10.31276240415595376760330310921, −9.351490264989220711870980104270, −8.417179031822736771246115005855, −7.995470520451125835280805836969, −5.72392395301681267383365889179, −4.63223001597295976816822739451, −3.63577046499708487394187885196, −0.39780999657034286067075317087, 2.18538687174989008575025256796, 3.97192763090095592200657455993, 5.87000946677899544026774846921, 7.17138244794621282223782927339, 7.81492667536668688292365436374, 8.753393491264351198571865563202, 10.18851438067977120230508544527, 10.84697910924127168956396280723, 12.18474619941949251887970880997, 12.88388924785711078573880783820

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