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Properties

Label 2-177-59.58-c8-0-24
Degree $2$
Conductor $177$
Sign $-0.999 - 0.00743i$
Analytic cond. $72.1060$
Root an. cond. $8.49152$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 29.4i·2-s − 46.7·3-s − 610.·4-s − 616.·5-s + 1.37e3i·6-s − 3.23e3·7-s + 1.04e4i·8-s + 2.18e3·9-s + 1.81e4i·10-s − 1.63e4i·11-s + 2.85e4·12-s + 2.25e4i·13-s + 9.51e4i·14-s + 2.88e4·15-s + 1.50e5·16-s − 1.18e4·17-s + ⋯
L(s)  = 1  − 1.83i·2-s − 0.577·3-s − 2.38·4-s − 0.986·5-s + 1.06i·6-s − 1.34·7-s + 2.54i·8-s + 0.333·9-s + 1.81i·10-s − 1.11i·11-s + 1.37·12-s + 0.789i·13-s + 2.47i·14-s + 0.569·15-s + 2.30·16-s − 0.141·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00743i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.999 - 0.00743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.999 - 0.00743i$
Analytic conductor: \(72.1060\)
Root analytic conductor: \(8.49152\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :4),\ -0.999 - 0.00743i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.3602865488\)
\(L(\frac12)\) \(\approx\) \(0.3602865488\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 46.7T \)
59 \( 1 + (1.21e7 + 9.00e4i)T \)
good2 \( 1 + 29.4iT - 256T^{2} \)
5 \( 1 + 616.T + 3.90e5T^{2} \)
7 \( 1 + 3.23e3T + 5.76e6T^{2} \)
11 \( 1 + 1.63e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.25e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.18e4T + 6.97e9T^{2} \)
19 \( 1 - 1.84e5T + 1.69e10T^{2} \)
23 \( 1 - 3.65e5iT - 7.83e10T^{2} \)
29 \( 1 - 3.33e5T + 5.00e11T^{2} \)
31 \( 1 + 7.30e5iT - 8.52e11T^{2} \)
37 \( 1 - 2.68e6iT - 3.51e12T^{2} \)
41 \( 1 + 5.13e6T + 7.98e12T^{2} \)
43 \( 1 - 1.15e6iT - 1.16e13T^{2} \)
47 \( 1 - 6.97e6iT - 2.38e13T^{2} \)
53 \( 1 + 1.05e7T + 6.22e13T^{2} \)
61 \( 1 + 1.31e7iT - 1.91e14T^{2} \)
67 \( 1 + 9.27e6iT - 4.06e14T^{2} \)
71 \( 1 + 6.40e6T + 6.45e14T^{2} \)
73 \( 1 + 1.74e7iT - 8.06e14T^{2} \)
79 \( 1 - 5.50e7T + 1.51e15T^{2} \)
83 \( 1 - 6.16e6iT - 2.25e15T^{2} \)
89 \( 1 + 7.65e7iT - 3.93e15T^{2} \)
97 \( 1 - 6.58e7iT - 7.83e15T^{2} \)
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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

−11.04853779643237235150661732306, −9.839839555809125237606830321628, −9.264513316607320571290444101500, −7.892836554241303096749077785911, −6.30925974321833265188376569236, −4.86688663756624627129534706443, −3.58190794619671044039234662761, −3.14872535482987016258246131508, −1.32981002395164370378501080269, −0.23986798694600225139777542810, 0.46238996452996245298326340618, 3.43998200232203434243518421228, 4.59519860989549826274906444034, 5.58675775915718893232068337453, 6.74947312634750401839592858394, 7.24388275375857796459514722896, 8.288827509224675440649000567638, 9.477916872045374072187174742243, 10.33102335224396444611947773324, 12.11132145142710159555175039165

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