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Properties

Label 2-171-171.103-c2-0-16
Degree $2$
Conductor $171$
Sign $0.991 + 0.127i$
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  + 3.78i·2-s + (−2.81 − 1.03i)3-s − 10.2·4-s + (−0.565 − 0.980i)5-s + (3.93 − 10.6i)6-s + (0.290 + 0.503i)7-s − 23.7i·8-s + (6.83 + 5.85i)9-s + (3.70 − 2.13i)10-s + (−5.05 − 8.76i)11-s + (28.9 + 10.7i)12-s − 2.42i·13-s + (−1.90 + 1.09i)14-s + (0.573 + 3.34i)15-s + 48.7·16-s + (8.04 − 13.9i)17-s + ⋯
L(s)  = 1  + 1.89i·2-s + (−0.938 − 0.346i)3-s − 2.57·4-s + (−0.113 − 0.196i)5-s + (0.655 − 1.77i)6-s + (0.0415 + 0.0719i)7-s − 2.97i·8-s + (0.759 + 0.650i)9-s + (0.370 − 0.213i)10-s + (−0.459 − 0.796i)11-s + (2.41 + 0.891i)12-s − 0.186i·13-s + (−0.136 + 0.0785i)14-s + (0.0382 + 0.223i)15-s + 3.04·16-s + (0.473 − 0.820i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.991 + 0.127i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.470607 - 0.0301268i\)
\(L(\frac12)\) \(\approx\) \(0.470607 - 0.0301268i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.81 + 1.03i)T \)
19 \( 1 + (-14.0 - 12.8i)T \)
good2 \( 1 - 3.78iT - 4T^{2} \)
5 \( 1 + (0.565 + 0.980i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-0.290 - 0.503i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (5.05 + 8.76i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 2.42iT - 169T^{2} \)
17 \( 1 + (-8.04 + 13.9i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 22.3T + 529T^{2} \)
29 \( 1 + (45.7 + 26.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (34.3 + 19.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 11.5iT - 1.36e3T^{2} \)
41 \( 1 + (-53.9 + 31.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 9.67T + 1.84e3T^{2} \)
47 \( 1 + (-6.28 + 10.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (30.2 - 17.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (19.4 - 11.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (34.3 - 59.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 122. iT - 4.48e3T^{2} \)
71 \( 1 + (119. + 69.2i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-2.55 + 4.43i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 54.7iT - 6.24e3T^{2} \)
83 \( 1 + (46.1 + 79.9i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (85.0 - 49.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 38.4iT - 9.40e3T^{2} \)
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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

−12.83525236881686293140185009946, −11.72189863223783761774086239341, −10.28024753004465616012965091505, −9.135417266103382864545118961624, −7.83886302803720637089144769922, −7.35592575288254760937832007280, −5.84629294258141533608797614610, −5.59349686499623499273924301407, −4.17183418618966908099471134752, −0.34503756538631495268663668969, 1.56629032939284051684100808825, 3.37733699521933774377419463686, 4.51439841072783713738845403182, 5.55964309703874890265989751459, 7.47596340863746524819338287889, 9.177779384330758941162702270201, 9.879207875719627149495587229867, 10.87443297311680509421206630924, 11.28306851145910043785606300113, 12.53203113280339662633452436525

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