Properties

Label 2-177-3.2-c2-0-26
Degree $2$
Conductor $177$
Sign $-0.266 + 0.963i$
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

Normalization:  

Dirichlet series

L(s)  = 1  + 1.90i·2-s + (−2.89 − 0.798i)3-s + 0.361·4-s − 0.0951i·5-s + (1.52 − 5.51i)6-s − 10.1·7-s + 8.31i·8-s + (7.72 + 4.61i)9-s + 0.181·10-s − 21.4i·11-s + (−1.04 − 0.288i)12-s − 18.5·13-s − 19.4i·14-s + (−0.0759 + 0.275i)15-s − 14.4·16-s − 11.7i·17-s + ⋯
L(s)  = 1  + 0.953i·2-s + (−0.963 − 0.266i)3-s + 0.0904·4-s − 0.0190i·5-s + (0.253 − 0.919i)6-s − 1.45·7-s + 1.03i·8-s + (0.858 + 0.512i)9-s + 0.0181·10-s − 1.95i·11-s + (−0.0872 − 0.0240i)12-s − 1.42·13-s − 1.38i·14-s + (−0.00506 + 0.0183i)15-s − 0.901·16-s − 0.693i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.266 + 0.963i$
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: \((2,\ 177,\ (\ :1),\ -0.266 + 0.963i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.124823 - 0.163948i\)
\(L(\frac12)\) \(\approx\) \(0.124823 - 0.163948i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.89 + 0.798i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 - 1.90iT - 4T^{2} \)
5 \( 1 + 0.0951iT - 25T^{2} \)
7 \( 1 + 10.1T + 49T^{2} \)
11 \( 1 + 21.4iT - 121T^{2} \)
13 \( 1 + 18.5T + 169T^{2} \)
17 \( 1 + 11.7iT - 289T^{2} \)
19 \( 1 + 16.0T + 361T^{2} \)
23 \( 1 + 4.05iT - 529T^{2} \)
29 \( 1 + 13.2iT - 841T^{2} \)
31 \( 1 + 10.0T + 961T^{2} \)
37 \( 1 + 26.6T + 1.36e3T^{2} \)
41 \( 1 - 42.3iT - 1.68e3T^{2} \)
43 \( 1 + 55.1T + 1.84e3T^{2} \)
47 \( 1 - 37.5iT - 2.20e3T^{2} \)
53 \( 1 - 72.8iT - 2.80e3T^{2} \)
61 \( 1 + 69.0T + 3.72e3T^{2} \)
67 \( 1 - 60.8T + 4.48e3T^{2} \)
71 \( 1 + 50.2iT - 5.04e3T^{2} \)
73 \( 1 - 29.1T + 5.32e3T^{2} \)
79 \( 1 - 59.8T + 6.24e3T^{2} \)
83 \( 1 + 92.7iT - 6.88e3T^{2} \)
89 \( 1 + 130. iT - 7.92e3T^{2} \)
97 \( 1 - 68.0T + 9.40e3T^{2} \)
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

−12.19238198854431941177691104370, −11.22959866087519419534994307091, −10.29143871388592307793119427940, −8.992613946280903790177092172389, −7.67012109297102517817085009132, −6.61746668998908275617717956153, −6.10891722408415797670826748163, −4.98995987793898946799763993474, −2.90595205766143857948790908884, −0.12655215050720986348108349976, 2.10174698888677841685914247541, 3.71790680433399748090253913650, 4.97407567528802970604104204933, 6.69136825875944731828397061522, 7.03522128823804514604788294736, 9.428941004416670804230599103019, 10.05065949939282733081930456263, 10.58618610592533508007512312576, 11.97576489305172458840976893367, 12.59200014940716121317148292707

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