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Properties

Label 2-171-171.110-c1-0-14
Degree $2$
Conductor $171$
Sign $0.670 + 0.742i$
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  + (1.98 − 1.66i)2-s + (0.915 + 1.47i)3-s + (0.819 − 4.64i)4-s + (−1.06 + 2.92i)5-s + (4.26 + 1.39i)6-s + (−1.52 − 2.64i)7-s + (−3.52 − 6.09i)8-s + (−1.32 + 2.69i)9-s + (2.76 + 7.58i)10-s − 3.42i·11-s + (7.57 − 3.05i)12-s + (0.864 + 2.37i)13-s + (−7.42 − 2.70i)14-s + (−5.27 + 1.11i)15-s + (−8.28 − 3.01i)16-s + (−1.79 + 4.92i)17-s + ⋯
L(s)  = 1  + (1.40 − 1.17i)2-s + (0.528 + 0.848i)3-s + (0.409 − 2.32i)4-s + (−0.476 + 1.30i)5-s + (1.74 + 0.568i)6-s + (−0.576 − 0.998i)7-s + (−1.24 − 2.15i)8-s + (−0.440 + 0.897i)9-s + (0.872 + 2.39i)10-s − 1.03i·11-s + (2.18 − 0.880i)12-s + (0.239 + 0.658i)13-s + (−1.98 − 0.722i)14-s + (−1.36 + 0.287i)15-s + (−2.07 − 0.754i)16-s + (−0.434 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.670 + 0.742i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.670 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08961 - 0.928351i\)
\(L(\frac12)\) \(\approx\) \(2.08961 - 0.928351i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.915 - 1.47i)T \)
19 \( 1 + (3.57 + 2.49i)T \)
good2 \( 1 + (-1.98 + 1.66i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.06 - 2.92i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.52 + 2.64i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.42iT - 11T^{2} \)
13 \( 1 + (-0.864 - 2.37i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.79 - 4.92i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.96 - 0.522i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.536 - 3.04i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + 0.794iT - 31T^{2} \)
37 \( 1 + 7.02iT - 37T^{2} \)
41 \( 1 + (-4.12 + 3.46i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.331 + 1.88i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-12.8 - 2.26i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-3.01 - 2.53i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.194 + 1.10i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (1.80 - 0.655i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.412 - 0.491i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (5.62 - 4.72i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.808 - 4.58i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.33 - 6.41i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (1.59 - 0.918i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.03 + 11.5i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (9.52 + 11.3i)T + (-16.8 + 95.5i)T^{2} \)
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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

−12.82198994229207188970906133883, −11.26557216166821284544930107685, −10.81843098249841656447365449014, −10.36178907271571852825500870562, −8.933921227648752327218134862112, −7.05865010962978537891688211138, −5.90620644586371051832012080784, −4.14104580933507858536697868680, −3.69063208915605254230255907950, −2.61315046738754447422858556097, 2.74769515233384640201927508964, 4.29519186936832943303082880676, 5.39051103503652240538669954350, 6.45522344545010131963087835525, 7.53538850035874915860778105243, 8.416519691851872697336088383125, 9.245841628767532322224238918102, 11.89388735658571010887104156842, 12.38060476896402524028946297027, 12.95898318351921147727940315993

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