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Properties

Label 2-171-19.12-c2-0-0
Degree $2$
Conductor $171$
Sign $-0.999 + 0.0226i$
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  + (−1.06 + 0.617i)2-s + (−1.23 + 2.14i)4-s + (4.08 + 7.08i)5-s − 10.4·7-s − 7.99i·8-s + (−8.73 − 5.04i)10-s + 3.90·11-s + (−2.02 − 1.16i)13-s + (11.1 − 6.46i)14-s + (−0.0227 − 0.0394i)16-s + (−2.13 − 3.70i)17-s + (−17.4 − 7.45i)19-s − 20.2·20-s + (−4.17 + 2.40i)22-s + (−20.0 + 34.7i)23-s + ⋯
L(s)  = 1  + (−0.534 + 0.308i)2-s + (−0.309 + 0.536i)4-s + (0.817 + 1.41i)5-s − 1.49·7-s − 0.999i·8-s + (−0.873 − 0.504i)10-s + 0.354·11-s + (−0.155 − 0.0898i)13-s + (0.799 − 0.461i)14-s + (−0.00142 − 0.00246i)16-s + (−0.125 − 0.217i)17-s + (−0.919 − 0.392i)19-s − 1.01·20-s + (−0.189 + 0.109i)22-s + (−0.872 + 1.51i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0226i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.999 + 0.0226i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ -0.999 + 0.0226i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00660751 - 0.584240i\)
\(L(\frac12)\) \(\approx\) \(0.00660751 - 0.584240i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (17.4 + 7.45i)T \)
good2 \( 1 + (1.06 - 0.617i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-4.08 - 7.08i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 10.4T + 49T^{2} \)
11 \( 1 - 3.90T + 121T^{2} \)
13 \( 1 + (2.02 + 1.16i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (2.13 + 3.70i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (20.0 - 34.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (10.6 + 6.17i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 22.2iT - 961T^{2} \)
37 \( 1 - 62.2iT - 1.36e3T^{2} \)
41 \( 1 + (-44.7 + 25.8i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-6.67 - 11.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (24.1 - 41.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-67.7 - 39.1i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (82.7 - 47.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-18.8 + 32.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (52.6 + 30.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-78.8 + 45.5i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-41.7 - 72.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-0.466 + 0.269i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 54.0T + 6.88e3T^{2} \)
89 \( 1 + (-81.6 - 47.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (126. - 73.2i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−13.26792979186973659899426805275, −12.09949393586050055489745615400, −10.74493810104503801831529742560, −9.700984970611877117871195465692, −9.350734304050331925641201678394, −7.71080741017069562013108333078, −6.72353193848163237240432637227, −6.09135242736894910363109220700, −3.79778835146608226904934236713, −2.72130761790985842943513473213, 0.40721440269600467823344194697, 2.05800577660206821932560172786, 4.26571464315779009862996505949, 5.60369901948174533915352557726, 6.44158181787951043739244953066, 8.449773041515736134688618132885, 9.144259367247533067808997578189, 9.827614984509967780167640870742, 10.63624310786190004125723550740, 12.33136155246207814018719961374

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