Properties

Label 2-171-171.110-c1-0-11
Degree $2$
Conductor $171$
Sign $0.0478 + 0.998i$
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  + (−0.788 + 0.661i)2-s + (−0.897 − 1.48i)3-s + (−0.163 + 0.926i)4-s + (0.446 − 1.22i)5-s + (1.68 + 0.573i)6-s + (−0.947 − 1.64i)7-s + (−1.51 − 2.62i)8-s + (−1.38 + 2.65i)9-s + (0.459 + 1.26i)10-s − 5.09i·11-s + (1.51 − 0.589i)12-s + (0.188 + 0.519i)13-s + (1.83 + 0.667i)14-s + (−2.21 + 0.439i)15-s + (1.16 + 0.422i)16-s + (1.79 − 4.93i)17-s + ⋯
L(s)  = 1  + (−0.557 + 0.467i)2-s + (−0.518 − 0.855i)3-s + (−0.0816 + 0.463i)4-s + (0.199 − 0.548i)5-s + (0.689 + 0.234i)6-s + (−0.358 − 0.620i)7-s + (−0.535 − 0.926i)8-s + (−0.462 + 0.886i)9-s + (0.145 + 0.399i)10-s − 1.53i·11-s + (0.438 − 0.170i)12-s + (0.0524 + 0.143i)13-s + (0.489 + 0.178i)14-s + (−0.572 + 0.113i)15-s + (0.290 + 0.105i)16-s + (0.435 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0478 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0478 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.0478 + 0.998i$
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: \((2,\ 171,\ (\ :1/2),\ 0.0478 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410071 - 0.390911i\)
\(L(\frac12)\) \(\approx\) \(0.410071 - 0.390911i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.897 + 1.48i)T \)
19 \( 1 + (1.28 + 4.16i)T \)
good2 \( 1 + (0.788 - 0.661i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (-0.446 + 1.22i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.947 + 1.64i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 5.09iT - 11T^{2} \)
13 \( 1 + (-0.188 - 0.519i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.79 + 4.93i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (4.43 + 0.782i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.744 - 4.22i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 - 9.56iT - 31T^{2} \)
37 \( 1 + 0.822iT - 37T^{2} \)
41 \( 1 + (4.69 - 3.93i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.620 + 3.51i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-6.74 - 1.18i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (3.87 + 3.24i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-1.14 - 6.46i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.20 + 0.802i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-10.0 + 11.9i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.276 + 0.232i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.920 - 5.22i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-4.77 + 13.1i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.96 + 4.02i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.457 - 2.59i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-1.47 - 1.76i)T + (-16.8 + 95.5i)T^{2} \)
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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

−12.57618392896998451014171650894, −11.66113604771792019826209381672, −10.57058723985070735451557349067, −9.107672703161608683076786655173, −8.377036023244943790138302390324, −7.25230345721752449214119570037, −6.48455519246557871149294797807, −5.14268969727741332785340805084, −3.23973167000580055434614263466, −0.67288756770921952253860439264, 2.23087434391682665482962086252, 4.09054965423608022739485485805, 5.58890798683969788481860365222, 6.34555671912650103874906495121, 8.172920636619565381165068252555, 9.517598909129448263426753367336, 10.00037201035597733253376745114, 10.67795812463845114854827108825, 11.83123267506606903612976569656, 12.57444394384049873267902669082

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