Properties

Label 2-171-171.121-c1-0-15
Degree $2$
Conductor $171$
Sign $-0.322 + 0.946i$
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  + (1.30 − 2.26i)2-s + (1.63 + 0.581i)3-s + (−2.40 − 4.16i)4-s − 2.00·5-s + (3.44 − 2.92i)6-s + (0.257 + 0.445i)7-s − 7.34·8-s + (2.32 + 1.89i)9-s + (−2.61 + 4.52i)10-s + (2.04 + 3.54i)11-s + (−1.50 − 8.20i)12-s + (−1.85 − 3.20i)13-s + 1.34·14-s + (−3.26 − 1.16i)15-s + (−4.77 + 8.26i)16-s + (3.60 + 6.24i)17-s + ⋯
L(s)  = 1  + (0.922 − 1.59i)2-s + (0.941 + 0.335i)3-s + (−1.20 − 2.08i)4-s − 0.895·5-s + (1.40 − 1.19i)6-s + (0.0971 + 0.168i)7-s − 2.59·8-s + (0.774 + 0.632i)9-s + (−0.826 + 1.43i)10-s + (0.617 + 1.06i)11-s + (−0.433 − 2.36i)12-s + (−0.513 − 0.889i)13-s + 0.358·14-s + (−0.843 − 0.300i)15-s + (−1.19 + 2.06i)16-s + (0.874 + 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.322 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.322 + 0.946i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.322 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13024 - 1.57870i\)
\(L(\frac12)\) \(\approx\) \(1.13024 - 1.57870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.63 - 0.581i)T \)
19 \( 1 + (-0.559 + 4.32i)T \)
good2 \( 1 + (-1.30 + 2.26i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 2.00T + 5T^{2} \)
7 \( 1 + (-0.257 - 0.445i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.04 - 3.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.85 + 3.20i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.60 - 6.24i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.174 + 0.301i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.54T + 29T^{2} \)
31 \( 1 + (0.773 - 1.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.82T + 37T^{2} \)
41 \( 1 + 2.92T + 41T^{2} \)
43 \( 1 + (0.200 - 0.347i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.16T + 47T^{2} \)
53 \( 1 + (-2.35 + 4.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.30T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + (0.480 + 0.831i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.26 + 5.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.31 - 2.28i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.55 - 6.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.37 + 14.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.10 - 8.84i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.64 + 16.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.50865762800835943760350577195, −11.66368162646693125200332750352, −10.51561076345526183009594925748, −9.840344546824306101598184633244, −8.721567436606042291032963778310, −7.41497213647203814424687766371, −5.28726544753036826555327950284, −4.13488854566990339812366628071, −3.39495402424805896350554164810, −1.94269311228105392462784841361, 3.39232090812837591909328876040, 4.17145746361551894541457160633, 5.67292024831985270013248481183, 7.06368290292929368560309996067, 7.56082407912426544499882075301, 8.513637892063063148817682054343, 9.445474123074456407349670619585, 11.68723526206542324716659938669, 12.33388863133746939144229819840, 13.63598290166328317982194770479

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