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Properties

Label 2-171-171.14-c1-0-17
Degree $2$
Conductor $171$
Sign $-0.904 - 0.427i$
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  + (−0.507 − 0.425i)2-s + (−1.51 − 0.832i)3-s + (−0.271 − 1.53i)4-s + (−0.735 − 2.02i)5-s + (0.415 + 1.06i)6-s + (−2.31 + 4.01i)7-s + (−1.17 + 2.04i)8-s + (1.61 + 2.52i)9-s + (−0.486 + 1.33i)10-s − 2.38i·11-s + (−0.869 + 2.56i)12-s + (−0.598 + 1.64i)13-s + (2.88 − 1.04i)14-s + (−0.565 + 3.68i)15-s + (−1.46 + 0.534i)16-s + (−1.52 − 4.20i)17-s + ⋯
L(s)  = 1  + (−0.358 − 0.300i)2-s + (−0.876 − 0.480i)3-s + (−0.135 − 0.769i)4-s + (−0.328 − 0.903i)5-s + (0.169 + 0.436i)6-s + (−0.876 + 1.51i)7-s + (−0.416 + 0.721i)8-s + (0.537 + 0.843i)9-s + (−0.153 + 0.422i)10-s − 0.720i·11-s + (−0.250 + 0.739i)12-s + (−0.166 + 0.456i)13-s + (0.770 − 0.280i)14-s + (−0.146 + 0.950i)15-s + (−0.367 + 0.133i)16-s + (−0.371 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.904 - 0.427i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ -0.904 - 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0500053 + 0.222898i\)
\(L(\frac12)\) \(\approx\) \(0.0500053 + 0.222898i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.51 + 0.832i)T \)
19 \( 1 + (3.09 + 3.06i)T \)
good2 \( 1 + (0.507 + 0.425i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (0.735 + 2.02i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (2.31 - 4.01i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.38iT - 11T^{2} \)
13 \( 1 + (0.598 - 1.64i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.52 + 4.20i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (5.67 - 1.00i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.13 + 6.45i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 - 3.03iT - 31T^{2} \)
37 \( 1 - 6.49iT - 37T^{2} \)
41 \( 1 + (0.680 + 0.571i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-1.88 + 10.6i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (8.21 - 1.44i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-5.26 + 4.41i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (1.08 - 6.15i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (1.43 + 0.522i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.95 - 2.33i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-7.57 - 6.35i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.89 + 10.7i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.738 + 2.02i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.85 + 1.07i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.28 + 12.9i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (1.18 - 1.41i)T + (-16.8 - 95.5i)T^{2} \)
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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

−11.88217056001961975798302717561, −11.56680559836443583782472038750, −10.15362516113676484707830311690, −9.151899124117597079995959280684, −8.424969043100346360019137830876, −6.57159465357558938842514282628, −5.73470235034208979860838964373, −4.79729359279182828579037937103, −2.26236862352009489043170146650, −0.25279823951764698610183187067, 3.54272778296965021255165864660, 4.22151814174084544462584274388, 6.33173703629456925097304138492, 6.99782392273872621653525728876, 7.924214665502561586167553494714, 9.606410118675746914926104291026, 10.35973831774959224310904334797, 11.06236297966465128246099138387, 12.49888036511188828495790140909, 12.93649882728978345241883725867

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