↑

Properties

Label 2-177-59.58-c6-0-45
Degree $2$
Conductor $177$
Sign $-0.0718 + 0.997i$
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  − 7.02i·2-s − 15.5·3-s + 14.5·4-s + 242.·5-s + 109. i·6-s + 243.·7-s − 552. i·8-s + 243·9-s − 1.70e3i·10-s − 1.55e3i·11-s − 227.·12-s + 1.41e3i·13-s − 1.70e3i·14-s − 3.77e3·15-s − 2.95e3·16-s + 3.19e3·17-s + ⋯
L(s)  = 1  − 0.878i·2-s − 0.577·3-s + 0.227·4-s + 1.93·5-s + 0.507i·6-s + 0.708·7-s − 1.07i·8-s + 0.333·9-s − 1.70i·10-s − 1.16i·11-s − 0.131·12-s + 0.644i·13-s − 0.622i·14-s − 1.11·15-s − 0.720·16-s + 0.650·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0718 + 0.997i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0718 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.0718 + 0.997i$
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: \((2,\ 177,\ (\ :3),\ -0.0718 + 0.997i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.246453603\)
\(L(\frac12)\) \(\approx\) \(3.246453603\)
\(L(4)\) 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 \( 1 + (-1.47e4 + 2.04e5i)T \)
good2 \( 1 + 7.02iT - 64T^{2} \)
5 \( 1 - 242.T + 1.56e4T^{2} \)
7 \( 1 - 243.T + 1.17e5T^{2} \)
11 \( 1 + 1.55e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.41e3iT - 4.82e6T^{2} \)
17 \( 1 - 3.19e3T + 2.41e7T^{2} \)
19 \( 1 + 454.T + 4.70e7T^{2} \)
23 \( 1 - 1.15e3iT - 1.48e8T^{2} \)
29 \( 1 - 2.19e4T + 5.94e8T^{2} \)
31 \( 1 - 4.41e3iT - 8.87e8T^{2} \)
37 \( 1 - 4.64e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.32e5T + 4.75e9T^{2} \)
43 \( 1 - 1.57e5iT - 6.32e9T^{2} \)
47 \( 1 + 1.63e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.92e5T + 2.21e10T^{2} \)
61 \( 1 - 3.40e5iT - 5.15e10T^{2} \)
67 \( 1 + 2.37e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.52e5T + 1.28e11T^{2} \)
73 \( 1 + 4.56e5iT - 1.51e11T^{2} \)
79 \( 1 - 3.30e4T + 2.43e11T^{2} \)
83 \( 1 - 4.05e4iT - 3.26e11T^{2} \)
89 \( 1 - 1.07e6iT - 4.96e11T^{2} \)
97 \( 1 + 1.45e6iT - 8.32e11T^{2} \)
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

−11.28496441331125941543164999366, −10.37722138049769716773749163584, −9.794116693489018200874429258620, −8.576594934491916086588543551462, −6.75845682922280008233803335377, −6.00097202429137472928497711133, −4.93512100030099735540931613060, −3.08376551955340178473268062464, −1.82564779707376268950552693861, −1.07900143105714814409316803945, 1.44934876389020193203645583585, 2.39579505533112721823581496671, 4.96886370778395413258612460032, 5.54921877994350561488167366584, 6.49579831406061594465636094923, 7.41029294497414348312280786837, 8.731780365286980185074328266015, 10.03006844587319747430361071247, 10.56696300026722853197194372295, 11.91178062421327490911460242221

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