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Properties

Label 2-177-3.2-c2-0-18
Degree $2$
Conductor $177$
Sign $0.574 - 0.818i$
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  − 0.00789i·2-s + (2.45 + 1.72i)3-s + 3.99·4-s + 4.78i·5-s + (0.0135 − 0.0193i)6-s + 0.326·7-s − 0.0631i·8-s + (3.06 + 8.46i)9-s + 0.0377·10-s − 7.17i·11-s + (9.82 + 6.88i)12-s − 19.3·13-s − 0.00257i·14-s + (−8.23 + 11.7i)15-s + 15.9·16-s − 10.2i·17-s + ⋯
L(s)  = 1  − 0.00394i·2-s + (0.818 + 0.574i)3-s + 0.999·4-s + 0.956i·5-s + (0.00226 − 0.00323i)6-s + 0.0466·7-s − 0.00789i·8-s + (0.340 + 0.940i)9-s + 0.00377·10-s − 0.652i·11-s + (0.818 + 0.574i)12-s − 1.49·13-s − 0.000184i·14-s + (−0.549 + 0.783i)15-s + 0.999·16-s − 0.604i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.574 - 0.818i$
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: \((2,\ 177,\ (\ :1),\ 0.574 - 0.818i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.01567 + 1.04859i\)
\(L(\frac12)\) \(\approx\) \(2.01567 + 1.04859i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.45 - 1.72i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 + 0.00789iT - 4T^{2} \)
5 \( 1 - 4.78iT - 25T^{2} \)
7 \( 1 - 0.326T + 49T^{2} \)
11 \( 1 + 7.17iT - 121T^{2} \)
13 \( 1 + 19.3T + 169T^{2} \)
17 \( 1 + 10.2iT - 289T^{2} \)
19 \( 1 + 1.08T + 361T^{2} \)
23 \( 1 + 32.0iT - 529T^{2} \)
29 \( 1 - 29.7iT - 841T^{2} \)
31 \( 1 - 41.9T + 961T^{2} \)
37 \( 1 - 15.1T + 1.36e3T^{2} \)
41 \( 1 - 12.5iT - 1.68e3T^{2} \)
43 \( 1 - 1.19T + 1.84e3T^{2} \)
47 \( 1 + 75.4iT - 2.20e3T^{2} \)
53 \( 1 + 69.0iT - 2.80e3T^{2} \)
61 \( 1 + 87.2T + 3.72e3T^{2} \)
67 \( 1 + 37.6T + 4.48e3T^{2} \)
71 \( 1 + 36.3iT - 5.04e3T^{2} \)
73 \( 1 - 25.2T + 5.32e3T^{2} \)
79 \( 1 + 103.T + 6.24e3T^{2} \)
83 \( 1 - 72.1iT - 6.88e3T^{2} \)
89 \( 1 - 143. iT - 7.92e3T^{2} \)
97 \( 1 - 57.9T + 9.40e3T^{2} \)
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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

−12.51728921316583034503027472351, −11.41666213784339581377133249599, −10.52758534825327718016512578697, −9.874609652228163153178198878024, −8.461263056359711894436592764426, −7.38968836052947549485298176226, −6.55734221323779859767868992592, −4.90168705823948325976758218326, −3.16091871865072433639752399002, −2.43739595142773507584724165150, 1.52021423766580669001977483106, 2.77690755069364862755021144164, 4.53444467775524968266860556816, 6.10234976611947593122996850244, 7.37044459540432324830286779782, 7.951685315212582566413029872131, 9.273637898419834468035163792692, 10.10489204308383734692135302087, 11.70042647962073186004081178452, 12.35853626652087038821744131723

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