Properties

Label 2-171-171.101-c2-0-23
Degree $2$
Conductor $171$
Sign $-0.372 + 0.928i$
Analytic cond. $4.65941$
Root an. cond. $2.15856$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.604 + 0.106i)2-s + (0.926 + 2.85i)3-s + (−3.40 + 1.23i)4-s + (−2.28 − 2.72i)5-s + (−0.863 − 1.62i)6-s + (−2.40 − 4.16i)7-s + (4.05 − 2.33i)8-s + (−7.28 + 5.28i)9-s + (1.67 + 1.40i)10-s − 6.82i·11-s + (−6.68 − 8.56i)12-s + (−5.99 − 5.03i)13-s + (1.89 + 2.26i)14-s + (5.65 − 9.04i)15-s + (8.90 − 7.47i)16-s + (−0.173 − 0.206i)17-s + ⋯
L(s)  = 1  + (−0.302 + 0.0532i)2-s + (0.308 + 0.951i)3-s + (−0.851 + 0.309i)4-s + (−0.457 − 0.544i)5-s + (−0.143 − 0.270i)6-s + (−0.343 − 0.594i)7-s + (0.506 − 0.292i)8-s + (−0.809 + 0.587i)9-s + (0.167 + 0.140i)10-s − 0.620i·11-s + (−0.557 − 0.713i)12-s + (−0.461 − 0.387i)13-s + (0.135 + 0.161i)14-s + (0.377 − 0.603i)15-s + (0.556 − 0.466i)16-s + (−0.0101 − 0.0121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.372 + 0.928i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ -0.372 + 0.928i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.171298 - 0.253338i\)
\(L(\frac12)\) \(\approx\) \(0.171298 - 0.253338i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.926 - 2.85i)T \)
19 \( 1 + (10.3 + 15.9i)T \)
good2 \( 1 + (0.604 - 0.106i)T + (3.75 - 1.36i)T^{2} \)
5 \( 1 + (2.28 + 2.72i)T + (-4.34 + 24.6i)T^{2} \)
7 \( 1 + (2.40 + 4.16i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + 6.82iT - 121T^{2} \)
13 \( 1 + (5.99 + 5.03i)T + (29.3 + 166. i)T^{2} \)
17 \( 1 + (0.173 + 0.206i)T + (-50.1 + 284. i)T^{2} \)
23 \( 1 + (-1.90 - 5.24i)T + (-405. + 340. i)T^{2} \)
29 \( 1 + (-6.89 - 18.9i)T + (-644. + 540. i)T^{2} \)
31 \( 1 + 53.3T + 961T^{2} \)
37 \( 1 + 46.1T + 1.36e3T^{2} \)
41 \( 1 + (37.9 - 6.68i)T + (1.57e3 - 574. i)T^{2} \)
43 \( 1 + (-10.9 - 3.97i)T + (1.41e3 + 1.18e3i)T^{2} \)
47 \( 1 + (-7.00 - 19.2i)T + (-1.69e3 + 1.41e3i)T^{2} \)
53 \( 1 + (-38.3 - 6.76i)T + (2.63e3 + 960. i)T^{2} \)
59 \( 1 + (32.6 - 89.6i)T + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (16.8 + 14.1i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (1.79 - 10.1i)T + (-4.21e3 - 1.53e3i)T^{2} \)
71 \( 1 + (-99.4 + 17.5i)T + (4.73e3 - 1.72e3i)T^{2} \)
73 \( 1 + (4.90 + 1.78i)T + (4.08e3 + 3.42e3i)T^{2} \)
79 \( 1 + (52.7 - 44.2i)T + (1.08e3 - 6.14e3i)T^{2} \)
83 \( 1 + (-75.1 + 43.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (53.3 + 146. i)T + (-6.06e3 + 5.09e3i)T^{2} \)
97 \( 1 + (14.4 + 82.0i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−12.31855517564624237654248215394, −10.92187120928148646045574048403, −10.10022428643662304547772722491, −9.015456151703591668007785852898, −8.474315264361135240696669189856, −7.29846174429366867781556928190, −5.31835945521443675930992462452, −4.32855239339496351223215517265, −3.32952145052800774052796961932, −0.19754635923477984723064285603, 1.98652972074755318719815122652, 3.70236285984355038478175572771, 5.39427511662700671666428559831, 6.71804938325833035109190789058, 7.73161404923847016688385036576, 8.755080970182703476462699871920, 9.612687287797127320179784758905, 10.81530537737752480939997147379, 12.09819501009175802716467209582, 12.73254113542152602672780924123

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