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Properties

Label 2-171-19.8-c2-0-1
Degree $2$
Conductor $171$
Sign $-0.710 - 0.703i$
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  + (−0.204 − 0.117i)2-s + (−1.97 − 3.41i)4-s + (−2.88 + 4.98i)5-s + 1.94·7-s + 1.87i·8-s + (1.17 − 0.678i)10-s − 8.46·11-s + (−16.7 + 9.66i)13-s + (−0.396 − 0.229i)14-s + (−7.66 + 13.2i)16-s + (−12.5 + 21.7i)17-s + (17.8 + 6.59i)19-s + 22.7·20-s + (1.72 + 0.997i)22-s + (−15.6 − 27.0i)23-s + ⋯
L(s)  = 1  + (−0.102 − 0.0588i)2-s + (−0.493 − 0.854i)4-s + (−0.576 + 0.997i)5-s + 0.277·7-s + 0.233i·8-s + (0.117 − 0.0678i)10-s − 0.769·11-s + (−1.28 + 0.743i)13-s + (−0.0283 − 0.0163i)14-s + (−0.479 + 0.830i)16-s + (−0.737 + 1.27i)17-s + (0.937 + 0.347i)19-s + 1.13·20-s + (0.0785 + 0.0453i)22-s + (−0.680 − 1.17i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.710 - 0.703i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ -0.710 - 0.703i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.151120 + 0.367496i\)
\(L(\frac12)\) \(\approx\) \(0.151120 + 0.367496i\)
\(L(2)\) 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 \( 1 + (-17.8 - 6.59i)T \)
good2 \( 1 + (0.204 + 0.117i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (2.88 - 4.98i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 1.94T + 49T^{2} \)
11 \( 1 + 8.46T + 121T^{2} \)
13 \( 1 + (16.7 - 9.66i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (12.5 - 21.7i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (15.6 + 27.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (13.7 - 7.91i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 14.4iT - 961T^{2} \)
37 \( 1 + 41.6iT - 1.36e3T^{2} \)
41 \( 1 + (1.53 + 0.885i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-14.4 + 25.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (1.70 + 2.95i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-80.0 + 46.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (4.27 + 2.46i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-7.45 - 12.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (70.4 - 40.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (52.9 + 30.5i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (38.8 - 67.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-84.0 - 48.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 102.T + 6.88e3T^{2} \)
89 \( 1 + (103. - 59.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (42.1 + 24.3i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.90901141687025967138010985357, −11.73868528310491390978038298765, −10.70217207068787500634594823953, −10.15957573108588634499717609469, −8.906649120795828907044252343545, −7.69696760117332475675606681253, −6.63114058705045403257828349859, −5.29969493777412332492728414974, −4.05993930815258456710308967975, −2.22734947434003077963034926984, 0.24161038665417289696026948509, 2.90844033227225179242095487131, 4.51786428977261461144157534958, 5.22378936050230622680252105843, 7.46112999790219348933278547240, 7.85592446818701754234219212072, 9.032432339798642021888118125195, 9.863455089780649066671900806674, 11.55102204976583797254777941732, 12.12334306928055431209374601632

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