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Properties

Label 2-177-59.58-c6-0-36
Degree $2$
Conductor $177$
Sign $-0.267 + 0.963i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.2i·2-s + 15.5·3-s − 112.·4-s + 130.·5-s − 206. i·6-s + 304.·7-s + 636. i·8-s + 243·9-s − 1.72e3i·10-s − 302. i·11-s − 1.74e3·12-s + 3.86e3i·13-s − 4.03e3i·14-s + 2.02e3·15-s + 1.28e3·16-s + 4.99e3·17-s + ⋯
L(s)  = 1  − 1.65i·2-s + 0.577·3-s − 1.75·4-s + 1.04·5-s − 0.957i·6-s + 0.887·7-s + 1.24i·8-s + 0.333·9-s − 1.72i·10-s − 0.227i·11-s − 1.01·12-s + 1.75i·13-s − 1.47i·14-s + 0.601·15-s + 0.312·16-s + 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.267 + 0.963i$
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: \((2,\ 177,\ (\ :3),\ -0.267 + 0.963i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.400902449\)
\(L(\frac12)\) \(\approx\) \(3.400902449\)
\(L(4)\) 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 \( 1 + (5.50e4 - 1.97e5i)T \)
good2 \( 1 + 13.2iT - 64T^{2} \)
5 \( 1 - 130.T + 1.56e4T^{2} \)
7 \( 1 - 304.T + 1.17e5T^{2} \)
11 \( 1 + 302. iT - 1.77e6T^{2} \)
13 \( 1 - 3.86e3iT - 4.82e6T^{2} \)
17 \( 1 - 4.99e3T + 2.41e7T^{2} \)
19 \( 1 - 1.17e4T + 4.70e7T^{2} \)
23 \( 1 - 1.70e4iT - 1.48e8T^{2} \)
29 \( 1 - 3.44e4T + 5.94e8T^{2} \)
31 \( 1 + 3.55e4iT - 8.87e8T^{2} \)
37 \( 1 + 3.28e4iT - 2.56e9T^{2} \)
41 \( 1 + 6.36e4T + 4.75e9T^{2} \)
43 \( 1 + 5.41e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.41e4iT - 1.07e10T^{2} \)
53 \( 1 - 8.64e4T + 2.21e10T^{2} \)
61 \( 1 - 2.18e4iT - 5.15e10T^{2} \)
67 \( 1 - 3.51e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.50e5T + 1.28e11T^{2} \)
73 \( 1 + 2.16e5iT - 1.51e11T^{2} \)
79 \( 1 + 7.97e5T + 2.43e11T^{2} \)
83 \( 1 + 7.10e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.09e6iT - 4.96e11T^{2} \)
97 \( 1 + 1.57e6iT - 8.32e11T^{2} \)
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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

−11.61288571524735485489368252228, −10.16710706333142374545951944442, −9.591219269766104508074375223219, −8.797714854590636109136556660281, −7.37178039408966571625467580293, −5.56947333764417282718974770022, −4.32629637997717033192198039726, −3.09653795757531311567468318882, −1.86760065682644095690287551678, −1.27410093986317685824608134068, 1.13222713907779725713451210488, 2.95501713359798729696504911090, 4.92783834585561920806701099138, 5.50307933808664600890213450617, 6.71530480722120479543214716941, 7.904992188023593864170825987522, 8.336213030802446029056835305683, 9.632297306212384222081894505244, 10.40828463017592966518369784020, 12.22780053770394259974572527463

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