Properties

Label 2-177-59.58-c2-0-6
Degree $2$
Conductor $177$
Sign $-0.499 - 0.866i$
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  + 2.72i·2-s − 1.73·3-s − 3.44·4-s + 5.71·5-s − 4.72i·6-s + 8.69·7-s + 1.52i·8-s + 2.99·9-s + 15.5i·10-s + 6.01i·11-s + 5.95·12-s − 4.81i·13-s + 23.7i·14-s − 9.89·15-s − 17.9·16-s + 10.8·17-s + ⋯
L(s)  = 1  + 1.36i·2-s − 0.577·3-s − 0.860·4-s + 1.14·5-s − 0.787i·6-s + 1.24·7-s + 0.190i·8-s + 0.333·9-s + 1.55i·10-s + 0.546i·11-s + 0.496·12-s − 0.370i·13-s + 1.69i·14-s − 0.659·15-s − 1.12·16-s + 0.637·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.499 - 0.866i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ -0.499 - 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.826459 + 1.43110i\)
\(L(\frac12)\) \(\approx\) \(0.826459 + 1.43110i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
59 \( 1 + (-29.4 - 51.1i)T \)
good2 \( 1 - 2.72iT - 4T^{2} \)
5 \( 1 - 5.71T + 25T^{2} \)
7 \( 1 - 8.69T + 49T^{2} \)
11 \( 1 - 6.01iT - 121T^{2} \)
13 \( 1 + 4.81iT - 169T^{2} \)
17 \( 1 - 10.8T + 289T^{2} \)
19 \( 1 + 18.4T + 361T^{2} \)
23 \( 1 - 15.4iT - 529T^{2} \)
29 \( 1 - 15.5T + 841T^{2} \)
31 \( 1 + 6.38iT - 961T^{2} \)
37 \( 1 + 0.527iT - 1.36e3T^{2} \)
41 \( 1 - 9.31T + 1.68e3T^{2} \)
43 \( 1 - 52.9iT - 1.84e3T^{2} \)
47 \( 1 + 78.0iT - 2.20e3T^{2} \)
53 \( 1 + 10.2T + 2.80e3T^{2} \)
61 \( 1 + 41.5iT - 3.72e3T^{2} \)
67 \( 1 + 92.4iT - 4.48e3T^{2} \)
71 \( 1 - 93.8T + 5.04e3T^{2} \)
73 \( 1 + 88.0iT - 5.32e3T^{2} \)
79 \( 1 + 81.2T + 6.24e3T^{2} \)
83 \( 1 + 2.92iT - 6.88e3T^{2} \)
89 \( 1 + 168. iT - 7.92e3T^{2} \)
97 \( 1 + 169. iT - 9.40e3T^{2} \)
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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

−13.06555125164158512117664437800, −11.78847878157488403737483749405, −10.73553997191787105615948129158, −9.675514736956391462975609473827, −8.408535147425674734088687678472, −7.49631242874829311250775902990, −6.34911328458730837372080835135, −5.50228858669751730570906226228, −4.69279653307237539847376860665, −1.89412007071514535499698581334, 1.25948377869023240248073928998, 2.39387978226828740447239763181, 4.23285799665214673766072422957, 5.44277750215200698952039499322, 6.65107589553893862308959890785, 8.354171272585797884540583211299, 9.523179583802051758295639080027, 10.48413884581424227833716455880, 11.05583063793301130196522063025, 11.96632530932837102648165981028

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