↑

Properties

Label 2-177-59.58-c6-0-11
Degree $2$
Conductor $177$
Sign $0.995 - 0.0902i$
Analytic cond. $40.7195$
Root an. cond. $6.38118$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 11.8i·2-s + 15.5·3-s − 77.4·4-s + 54.6·5-s − 185. i·6-s − 559.·7-s + 159. i·8-s + 243·9-s − 649. i·10-s + 2.29e3i·11-s − 1.20e3·12-s − 2.22e3i·13-s + 6.65e3i·14-s + 851.·15-s − 3.05e3·16-s − 3.18e3·17-s + ⋯
L(s)  = 1  − 1.48i·2-s + 0.577·3-s − 1.20·4-s + 0.436·5-s − 0.858i·6-s − 1.63·7-s + 0.311i·8-s + 0.333·9-s − 0.649i·10-s + 1.72i·11-s − 0.698·12-s − 1.01i·13-s + 2.42i·14-s + 0.252·15-s − 0.746·16-s − 0.648·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0902i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.995 - 0.0902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.995 - 0.0902i$
Analytic conductor: \(40.7195\)
Root analytic conductor: \(6.38118\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3),\ 0.995 - 0.0902i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.212092404\)
\(L(\frac12)\) \(\approx\) \(1.212092404\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 15.5T \)
59 \( 1 + (-2.04e5 + 1.85e4i)T \)
good2 \( 1 + 11.8iT - 64T^{2} \)
5 \( 1 - 54.6T + 1.56e4T^{2} \)
7 \( 1 + 559.T + 1.17e5T^{2} \)
11 \( 1 - 2.29e3iT - 1.77e6T^{2} \)
13 \( 1 + 2.22e3iT - 4.82e6T^{2} \)
17 \( 1 + 3.18e3T + 2.41e7T^{2} \)
19 \( 1 - 7.58e3T + 4.70e7T^{2} \)
23 \( 1 - 1.10e4iT - 1.48e8T^{2} \)
29 \( 1 - 3.28e4T + 5.94e8T^{2} \)
31 \( 1 - 9.81e3iT - 8.87e8T^{2} \)
37 \( 1 - 4.86e4iT - 2.56e9T^{2} \)
41 \( 1 - 4.47e4T + 4.75e9T^{2} \)
43 \( 1 - 1.40e5iT - 6.32e9T^{2} \)
47 \( 1 - 1.38e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.27e5T + 2.21e10T^{2} \)
61 \( 1 + 2.91e4iT - 5.15e10T^{2} \)
67 \( 1 + 4.01e5iT - 9.04e10T^{2} \)
71 \( 1 + 6.97e5T + 1.28e11T^{2} \)
73 \( 1 + 2.09e5iT - 1.51e11T^{2} \)
79 \( 1 + 4.57e5T + 2.43e11T^{2} \)
83 \( 1 - 2.67e5iT - 3.26e11T^{2} \)
89 \( 1 - 2.20e5iT - 4.96e11T^{2} \)
97 \( 1 + 9.55e5iT - 8.32e11T^{2} \)
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

−11.73874794443677688339173506265, −10.27041599700363281762463886647, −9.791298318037061532160626971026, −9.284532039250353834139143937555, −7.55724240081793278488151081107, −6.37409132394367878129857376366, −4.62720532005380467882203599726, −3.30107049916325257441428352718, −2.61007539465935291713133287365, −1.29645498731776052157961830962, 0.32108349404368630070942043600, 2.63745963187420601258251099951, 3.94653929437010704341911669207, 5.65309083284667287866199386848, 6.39760484293004996512106165900, 7.17860975618311655613282645565, 8.597539725370751548195317681909, 9.084989067649565651897892368839, 10.16933538537932858680908515459, 11.66385972133103525922435098684

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