Properties

Label 2-171-171.101-c2-0-3
Degree $2$
Conductor $171$
Sign $-0.332 - 0.943i$
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  + (−2.68 + 0.473i)2-s + (−2.95 + 0.527i)3-s + (3.22 − 1.17i)4-s + (4.68 + 5.58i)5-s + (7.67 − 2.81i)6-s + (−2.46 − 4.26i)7-s + (1.34 − 0.778i)8-s + (8.44 − 3.11i)9-s + (−15.2 − 12.7i)10-s − 7.13i·11-s + (−8.89 + 5.16i)12-s + (7.90 + 6.63i)13-s + (8.62 + 10.2i)14-s + (−16.7 − 14.0i)15-s + (−13.7 + 11.5i)16-s + (18.5 + 22.0i)17-s + ⋯
L(s)  = 1  + (−1.34 + 0.236i)2-s + (−0.984 + 0.175i)3-s + (0.805 − 0.293i)4-s + (0.936 + 1.11i)5-s + (1.27 − 0.469i)6-s + (−0.351 − 0.608i)7-s + (0.168 − 0.0973i)8-s + (0.938 − 0.346i)9-s + (−1.52 − 1.27i)10-s − 0.648i·11-s + (−0.741 + 0.430i)12-s + (0.608 + 0.510i)13-s + (0.615 + 0.733i)14-s + (−1.11 − 0.933i)15-s + (−0.859 + 0.721i)16-s + (1.08 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.332 - 0.943i$
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: \((2,\ 171,\ (\ :1),\ -0.332 - 0.943i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.304672 + 0.430548i\)
\(L(\frac12)\) \(\approx\) \(0.304672 + 0.430548i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.95 - 0.527i)T \)
19 \( 1 + (18.9 - 0.0106i)T \)
good2 \( 1 + (2.68 - 0.473i)T + (3.75 - 1.36i)T^{2} \)
5 \( 1 + (-4.68 - 5.58i)T + (-4.34 + 24.6i)T^{2} \)
7 \( 1 + (2.46 + 4.26i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + 7.13iT - 121T^{2} \)
13 \( 1 + (-7.90 - 6.63i)T + (29.3 + 166. i)T^{2} \)
17 \( 1 + (-18.5 - 22.0i)T + (-50.1 + 284. i)T^{2} \)
23 \( 1 + (-7.46 - 20.5i)T + (-405. + 340. i)T^{2} \)
29 \( 1 + (-2.84 - 7.81i)T + (-644. + 540. i)T^{2} \)
31 \( 1 + 31.7T + 961T^{2} \)
37 \( 1 + 14.6T + 1.36e3T^{2} \)
41 \( 1 + (41.1 - 7.26i)T + (1.57e3 - 574. i)T^{2} \)
43 \( 1 + (-61.1 - 22.2i)T + (1.41e3 + 1.18e3i)T^{2} \)
47 \( 1 + (-23.6 - 64.9i)T + (-1.69e3 + 1.41e3i)T^{2} \)
53 \( 1 + (43.7 + 7.71i)T + (2.63e3 + 960. i)T^{2} \)
59 \( 1 + (21.0 - 57.9i)T + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (-43.1 - 36.1i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (6.23 - 35.3i)T + (-4.21e3 - 1.53e3i)T^{2} \)
71 \( 1 + (-39.6 + 6.99i)T + (4.73e3 - 1.72e3i)T^{2} \)
73 \( 1 + (56.6 + 20.6i)T + (4.08e3 + 3.42e3i)T^{2} \)
79 \( 1 + (-82.4 + 69.1i)T + (1.08e3 - 6.14e3i)T^{2} \)
83 \( 1 + (61.2 - 35.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-19.1 - 52.5i)T + (-6.06e3 + 5.09e3i)T^{2} \)
97 \( 1 + (9.63 + 54.6i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−12.80661395169578611991972244009, −11.15659144042338090531230991849, −10.59581304758273944461405335239, −10.03842118549825114481915263750, −8.999014446820748672129719913113, −7.53975431301833676624976724021, −6.55911641593234931996769941206, −5.88916978153591398061446582156, −3.79268926481214292886516402263, −1.42904875784079428916030673770, 0.62131639840859696138665275066, 1.98280968963931956442013184348, 4.87407596767268453233700348715, 5.74984665983971706171155602659, 7.10360811679720904067814639335, 8.433742294964248988756110316453, 9.343945915499801769534803063436, 10.02732599381832542361248355163, 10.95491779956213013773742835883, 12.23197141272811162643667291006

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