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Properties

Label 2-171-171.14-c1-0-8
Degree $2$
Conductor $171$
Sign $0.600 + 0.799i$
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  + (0.103 + 0.0865i)2-s + (−1.73 − 0.0351i)3-s + (−0.344 − 1.95i)4-s + (0.862 + 2.36i)5-s + (−0.175 − 0.153i)6-s + (1.62 − 2.81i)7-s + (0.268 − 0.464i)8-s + (2.99 + 0.121i)9-s + (−0.116 + 0.319i)10-s − 5.59i·11-s + (0.527 + 3.39i)12-s + (1.44 − 3.97i)13-s + (0.411 − 0.149i)14-s + (−1.41 − 4.13i)15-s + (−3.65 + 1.33i)16-s + (1.48 + 4.06i)17-s + ⋯
L(s)  = 1  + (0.0729 + 0.0612i)2-s + (−0.999 − 0.0202i)3-s + (−0.172 − 0.975i)4-s + (0.385 + 1.05i)5-s + (−0.0717 − 0.0626i)6-s + (0.614 − 1.06i)7-s + (0.0948 − 0.164i)8-s + (0.999 + 0.0405i)9-s + (−0.0367 + 0.100i)10-s − 1.68i·11-s + (0.152 + 0.979i)12-s + (0.401 − 1.10i)13-s + (0.110 − 0.0400i)14-s + (−0.364 − 1.06i)15-s + (−0.914 + 0.332i)16-s + (0.359 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.600 + 0.799i$
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: \((2,\ 171,\ (\ :1/2),\ 0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.853076 - 0.426121i\)
\(L(\frac12)\) \(\approx\) \(0.853076 - 0.426121i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0351i)T \)
19 \( 1 + (2.66 - 3.45i)T \)
good2 \( 1 + (-0.103 - 0.0865i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (-0.862 - 2.36i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-1.62 + 2.81i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 5.59iT - 11T^{2} \)
13 \( 1 + (-1.44 + 3.97i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.48 - 4.06i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-3.14 + 0.554i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.06 - 6.05i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 - 6.64iT - 31T^{2} \)
37 \( 1 + 4.71iT - 37T^{2} \)
41 \( 1 + (4.53 + 3.80i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-1.19 + 6.77i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.139 - 0.0246i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (4.63 - 3.88i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (0.831 - 4.71i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-4.55 - 1.65i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.35 + 2.80i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-4.47 - 3.75i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.71 - 9.70i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-3.30 - 9.08i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-3.95 - 2.28i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.35 - 13.3i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-2.21 + 2.64i)T + (-16.8 - 95.5i)T^{2} \)
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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

−12.68874182118847228972473107447, −11.01068785971589389247590711484, −10.66195937893256648997069021151, −10.33064115197518451142814316320, −8.474538544338503842764112187155, −7.01847641538986716075435833438, −6.08766825586346717298849434547, −5.33595129012376669822313569451, −3.68606435196576836901226659154, −1.13272943573205675730810335301, 2.02910922901957181974887203325, 4.59689968263776523109787853019, 4.90615195767766992618205294305, 6.52564941947752605607136200784, 7.73652397837672265756962730377, 9.024143523869415339182421596727, 9.638682981391082061588235583616, 11.52214407614818526084355071885, 11.82612965046451795851694221336, 12.76455986299687220795803393805

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