Properties

Label 2-177-3.2-c2-0-28
Degree $2$
Conductor $177$
Sign $0.204 + 0.978i$
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.50i·2-s + (2.93 − 0.614i)3-s + 1.73·4-s + 1.04i·5-s + (−0.924 − 4.41i)6-s − 6.70·7-s − 8.63i·8-s + (8.24 − 3.60i)9-s + 1.57·10-s − 13.0i·11-s + (5.10 − 1.06i)12-s + 15.6·13-s + 10.0i·14-s + (0.643 + 3.07i)15-s − 6.03·16-s + 23.4i·17-s + ⋯
L(s)  = 1  − 0.752i·2-s + (0.978 − 0.204i)3-s + 0.434·4-s + 0.209i·5-s + (−0.154 − 0.736i)6-s − 0.958·7-s − 1.07i·8-s + (0.916 − 0.400i)9-s + 0.157·10-s − 1.18i·11-s + (0.425 − 0.0889i)12-s + 1.20·13-s + 0.720i·14-s + (0.0428 + 0.204i)15-s − 0.377·16-s + 1.37i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.204 + 0.978i$
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: \((2,\ 177,\ (\ :1),\ 0.204 + 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.72172 - 1.39878i\)
\(L(\frac12)\) \(\approx\) \(1.72172 - 1.39878i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.93 + 0.614i)T \)
59 \( 1 + 7.68iT \)
good2 \( 1 + 1.50iT - 4T^{2} \)
5 \( 1 - 1.04iT - 25T^{2} \)
7 \( 1 + 6.70T + 49T^{2} \)
11 \( 1 + 13.0iT - 121T^{2} \)
13 \( 1 - 15.6T + 169T^{2} \)
17 \( 1 - 23.4iT - 289T^{2} \)
19 \( 1 + 3.66T + 361T^{2} \)
23 \( 1 + 7.48iT - 529T^{2} \)
29 \( 1 - 55.2iT - 841T^{2} \)
31 \( 1 + 48.1T + 961T^{2} \)
37 \( 1 + 61.5T + 1.36e3T^{2} \)
41 \( 1 - 23.5iT - 1.68e3T^{2} \)
43 \( 1 + 2.28T + 1.84e3T^{2} \)
47 \( 1 - 89.1iT - 2.20e3T^{2} \)
53 \( 1 - 11.9iT - 2.80e3T^{2} \)
61 \( 1 - 69.8T + 3.72e3T^{2} \)
67 \( 1 + 41.2T + 4.48e3T^{2} \)
71 \( 1 + 73.6iT - 5.04e3T^{2} \)
73 \( 1 - 78.0T + 5.32e3T^{2} \)
79 \( 1 - 16.5T + 6.24e3T^{2} \)
83 \( 1 - 17.9iT - 6.88e3T^{2} \)
89 \( 1 - 11.3iT - 7.92e3T^{2} \)
97 \( 1 - 29.9T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.62470105355517067146639875776, −10.95303757412820265863338945720, −10.54424394333686489567900884716, −9.192788386609629547440675136476, −8.411495031269216008365806485993, −6.93814962219247498822649723472, −6.15121448856678992175965970649, −3.61941466669022576532189733386, −3.16635143384651288416116940065, −1.47185661930339370980343650217, 2.20323221454575586901405104976, 3.63846169988001091514758533505, 5.20967330699460365135846160777, 6.70430373212110727353835038542, 7.35825561909100931530824588386, 8.559915299770506103171604272557, 9.444186104538716919723194671927, 10.45329312132958158997043484686, 11.78212907847139098876595330533, 12.94999199002312446991624130474

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