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Properties

Label 2-177-3.2-c2-0-27
Degree $2$
Conductor $177$
Sign $-0.558 + 0.829i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
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  − 1.30i·2-s + (−2.48 − 1.67i)3-s + 2.29·4-s − 3.80i·5-s + (−2.18 + 3.25i)6-s + 12.3·7-s − 8.22i·8-s + (3.38 + 8.33i)9-s − 4.97·10-s − 1.19i·11-s + (−5.70 − 3.83i)12-s − 14.6·13-s − 16.1i·14-s + (−6.37 + 9.47i)15-s − 1.58·16-s − 23.0i·17-s + ⋯
L(s)  = 1  − 0.653i·2-s + (−0.829 − 0.558i)3-s + 0.572·4-s − 0.761i·5-s + (−0.364 + 0.542i)6-s + 1.76·7-s − 1.02i·8-s + (0.376 + 0.926i)9-s − 0.497·10-s − 0.108i·11-s + (−0.475 − 0.319i)12-s − 1.12·13-s − 1.15i·14-s + (−0.425 + 0.631i)15-s − 0.0990·16-s − 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.558 + 0.829i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ -0.558 + 0.829i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.703899 - 1.32217i\)
\(L(\frac12)\) \(\approx\) \(0.703899 - 1.32217i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.48 + 1.67i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 + 1.30iT - 4T^{2} \)
5 \( 1 + 3.80iT - 25T^{2} \)
7 \( 1 - 12.3T + 49T^{2} \)
11 \( 1 + 1.19iT - 121T^{2} \)
13 \( 1 + 14.6T + 169T^{2} \)
17 \( 1 + 23.0iT - 289T^{2} \)
19 \( 1 + 14.4T + 361T^{2} \)
23 \( 1 - 9.85iT - 529T^{2} \)
29 \( 1 - 46.0iT - 841T^{2} \)
31 \( 1 + 19.2T + 961T^{2} \)
37 \( 1 + 3.90T + 1.36e3T^{2} \)
41 \( 1 + 28.4iT - 1.68e3T^{2} \)
43 \( 1 - 57.0T + 1.84e3T^{2} \)
47 \( 1 - 42.9iT - 2.20e3T^{2} \)
53 \( 1 - 47.5iT - 2.80e3T^{2} \)
61 \( 1 - 39.9T + 3.72e3T^{2} \)
67 \( 1 + 54.3T + 4.48e3T^{2} \)
71 \( 1 + 79.8iT - 5.04e3T^{2} \)
73 \( 1 - 123.T + 5.32e3T^{2} \)
79 \( 1 - 28.8T + 6.24e3T^{2} \)
83 \( 1 + 2.13iT - 6.88e3T^{2} \)
89 \( 1 - 161. iT - 7.92e3T^{2} \)
97 \( 1 - 0.232T + 9.40e3T^{2} \)
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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

−12.16802125019704504840161748544, −11.17036544526857948432388603589, −10.69003877630178327332508051640, −9.173206744325804214263786404583, −7.77009075260332208663694328774, −7.04745839328915308437290147038, −5.38515054986606822341529249937, −4.63087806753802337114633637753, −2.24970025808025093434902485536, −1.07121095489713826863885030111, 2.13357013928372468775894784519, 4.33217288753122564671060709209, 5.40718729878314093889501834029, 6.45637717820514848779619574884, 7.48249201746881382490427886692, 8.445665841012758991192244113343, 10.18256466472173573477263212458, 10.93129303587878363161838272472, 11.51491905634548973791424648275, 12.49204418357667530977182581106

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