Properties

Label 2-177-3.2-c2-0-29
Degree $2$
Conductor $177$
Sign $-0.987 - 0.158i$
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  − 2.97i·2-s + (−0.474 + 2.96i)3-s − 4.85·4-s − 3.00i·5-s + (8.81 + 1.41i)6-s − 3.67·7-s + 2.55i·8-s + (−8.54 − 2.81i)9-s − 8.94·10-s − 14.3i·11-s + (2.30 − 14.3i)12-s − 8.01·13-s + 10.9i·14-s + (8.90 + 1.42i)15-s − 11.8·16-s − 0.828i·17-s + ⋯
L(s)  = 1  − 1.48i·2-s + (−0.158 + 0.987i)3-s − 1.21·4-s − 0.601i·5-s + (1.46 + 0.235i)6-s − 0.525·7-s + 0.319i·8-s + (−0.949 − 0.312i)9-s − 0.894·10-s − 1.30i·11-s + (0.192 − 1.19i)12-s − 0.616·13-s + 0.782i·14-s + (0.593 + 0.0951i)15-s − 0.739·16-s − 0.0487i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.987 - 0.158i$
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: \((2,\ 177,\ (\ :1),\ -0.987 - 0.158i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0605743 + 0.760688i\)
\(L(\frac12)\) \(\approx\) \(0.0605743 + 0.760688i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.474 - 2.96i)T \)
59 \( 1 + 7.68iT \)
good2 \( 1 + 2.97iT - 4T^{2} \)
5 \( 1 + 3.00iT - 25T^{2} \)
7 \( 1 + 3.67T + 49T^{2} \)
11 \( 1 + 14.3iT - 121T^{2} \)
13 \( 1 + 8.01T + 169T^{2} \)
17 \( 1 + 0.828iT - 289T^{2} \)
19 \( 1 + 25.2T + 361T^{2} \)
23 \( 1 + 7.83iT - 529T^{2} \)
29 \( 1 - 2.34iT - 841T^{2} \)
31 \( 1 - 25.4T + 961T^{2} \)
37 \( 1 + 0.422T + 1.36e3T^{2} \)
41 \( 1 + 6.28iT - 1.68e3T^{2} \)
43 \( 1 - 48.4T + 1.84e3T^{2} \)
47 \( 1 + 63.1iT - 2.20e3T^{2} \)
53 \( 1 - 24.6iT - 2.80e3T^{2} \)
61 \( 1 - 73.6T + 3.72e3T^{2} \)
67 \( 1 + 100.T + 4.48e3T^{2} \)
71 \( 1 + 47.9iT - 5.04e3T^{2} \)
73 \( 1 - 130.T + 5.32e3T^{2} \)
79 \( 1 + 76.0T + 6.24e3T^{2} \)
83 \( 1 + 142. iT - 6.88e3T^{2} \)
89 \( 1 - 73.4iT - 7.92e3T^{2} \)
97 \( 1 - 84.6T + 9.40e3T^{2} \)
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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

−11.80297106107553639254317414455, −10.85021776480335463447202641572, −10.23083618223670774313342914269, −9.188924226912385438965094072230, −8.522946106869642847472270537503, −6.32428249889824380664428925610, −4.90310215219201926220468544187, −3.82701923130854466453054817910, −2.69732905977761190226735165472, −0.44607198403415218685295639086, 2.43139976637656382833909849441, 4.68402875119699069604619588504, 6.05134840998559125417098685960, 6.82265665342073600474998456198, 7.42422597692704693681265563257, 8.462478430323792589534220894018, 9.727328620440980442938539606704, 11.05663142652024460867142561689, 12.36476677066018199045201116201, 13.06458642522607676791569507670

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