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Properties

Label 2-171-171.103-c2-0-15
Degree $2$
Conductor $171$
Sign $0.993 + 0.114i$
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  − 1.77i·2-s + (1.70 + 2.47i)3-s + 0.859·4-s + (−0.983 − 1.70i)5-s + (4.37 − 3.01i)6-s + (6.71 + 11.6i)7-s − 8.61i·8-s + (−3.21 + 8.40i)9-s + (−3.01 + 1.74i)10-s + (−0.278 − 0.482i)11-s + (1.46 + 2.12i)12-s − 1.13i·13-s + (20.6 − 11.8i)14-s + (2.53 − 5.32i)15-s − 11.8·16-s + (−3.70 + 6.41i)17-s + ⋯
L(s)  = 1  − 0.886i·2-s + (0.566 + 0.823i)3-s + 0.214·4-s + (−0.196 − 0.340i)5-s + (0.729 − 0.502i)6-s + (0.958 + 1.66i)7-s − 1.07i·8-s + (−0.357 + 0.934i)9-s + (−0.301 + 0.174i)10-s + (−0.0253 − 0.0438i)11-s + (0.121 + 0.176i)12-s − 0.0875i·13-s + (1.47 − 0.849i)14-s + (0.169 − 0.355i)15-s − 0.739·16-s + (−0.217 + 0.377i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.993 + 0.114i$
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: \((2,\ 171,\ (\ :1),\ 0.993 + 0.114i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.05472 - 0.117768i\)
\(L(\frac12)\) \(\approx\) \(2.05472 - 0.117768i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 2.47i)T \)
19 \( 1 + (-16.6 + 9.14i)T \)
good2 \( 1 + 1.77iT - 4T^{2} \)
5 \( 1 + (0.983 + 1.70i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-6.71 - 11.6i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (0.278 + 0.482i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 1.13iT - 169T^{2} \)
17 \( 1 + (3.70 - 6.41i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 23.2T + 529T^{2} \)
29 \( 1 + (24.5 + 14.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (36.5 + 21.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 7.57iT - 1.36e3T^{2} \)
41 \( 1 + (-11.0 + 6.39i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 13.1T + 1.84e3T^{2} \)
47 \( 1 + (-31.7 + 54.9i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (67.9 - 39.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (88.1 - 50.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (8.81 - 15.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + 23.8iT - 4.48e3T^{2} \)
71 \( 1 + (-36.5 - 21.0i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (25.3 - 43.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 121. iT - 6.24e3T^{2} \)
83 \( 1 + (-5.13 - 8.88i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-101. + 58.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 88.5iT - 9.40e3T^{2} \)
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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

−12.23144402145512182283882152629, −11.42705519628236858596736905323, −10.74510802235189749098405706852, −9.394707957543142513983315278786, −8.820285784419456460726184168100, −7.64578190518933413037231872536, −5.71509251884973771560219646866, −4.59211403261819938220296693057, −3.07836094259892726347084414040, −1.99666127172658042638947774706, 1.49349017557516466267530603238, 3.38852668056862035975469303881, 5.09696979195328930773109589403, 6.68752025761167767290692127992, 7.42932472403129007274646428625, 7.80769831180512170173698971594, 9.164524890310853010662820096364, 10.86077942151115068759636617736, 11.36033341949792700556581634132, 12.78840854884838248693584302813

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