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Properties

Label 2-177-177.176-c5-0-25
Degree $2$
Conductor $177$
Sign $-0.625 - 0.779i$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.62·2-s + (−4.57 + 14.9i)3-s + 42.3·4-s + 15.6i·5-s + (39.4 − 128. i)6-s − 96.1·7-s − 89.3·8-s + (−201. − 136. i)9-s − 134. i·10-s + 350.·11-s + (−193. + 631. i)12-s + 274. i·13-s + 829.·14-s + (−233. − 71.5i)15-s − 585.·16-s + 852. i·17-s + ⋯
L(s)  = 1  − 1.52·2-s + (−0.293 + 0.956i)3-s + 1.32·4-s + 0.279i·5-s + (0.447 − 1.45i)6-s − 0.741·7-s − 0.493·8-s + (−0.827 − 0.560i)9-s − 0.426i·10-s + 0.872·11-s + (−0.388 + 1.26i)12-s + 0.450i·13-s + 1.13·14-s + (−0.267 − 0.0821i)15-s − 0.571·16-s + 0.715i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.779i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.625 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.625 - 0.779i$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -0.625 - 0.779i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6201567700\)
\(L(\frac12)\) \(\approx\) \(0.6201567700\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.57 - 14.9i)T \)
59 \( 1 + (-1.50e4 + 2.21e4i)T \)
good2 \( 1 + 8.62T + 32T^{2} \)
5 \( 1 - 15.6iT - 3.12e3T^{2} \)
7 \( 1 + 96.1T + 1.68e4T^{2} \)
11 \( 1 - 350.T + 1.61e5T^{2} \)
13 \( 1 - 274. iT - 3.71e5T^{2} \)
17 \( 1 - 852. iT - 1.41e6T^{2} \)
19 \( 1 - 764.T + 2.47e6T^{2} \)
23 \( 1 - 4.62e3T + 6.43e6T^{2} \)
29 \( 1 + 4.30e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.77e3iT - 2.86e7T^{2} \)
37 \( 1 + 7.39e3iT - 6.93e7T^{2} \)
41 \( 1 - 2.07e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.45e4iT - 1.47e8T^{2} \)
47 \( 1 - 9.94e3T + 2.29e8T^{2} \)
53 \( 1 - 1.88e4iT - 4.18e8T^{2} \)
61 \( 1 - 2.33e3iT - 8.44e8T^{2} \)
67 \( 1 + 4.24e3iT - 1.35e9T^{2} \)
71 \( 1 - 6.97e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.48e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.32e4T + 3.07e9T^{2} \)
83 \( 1 + 1.09e5T + 3.93e9T^{2} \)
89 \( 1 + 8.38e4T + 5.58e9T^{2} \)
97 \( 1 + 2.75e4iT - 8.58e9T^{2} \)
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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

−11.49358313873297956185418675949, −10.95903592641257927027004986709, −9.840107535007430347695602339308, −9.371298233791769095208193855053, −8.493631231781905139598783351302, −7.05286093829047949809027073310, −6.15160395536269961099630443416, −4.41209983866250706445990830417, −2.96434739390730257485267753226, −1.00820362804457127421849167309, 0.47674757361451063165094560379, 1.33535881056254689923756681265, 2.94963432069157025073700375740, 5.25560912331202349967518049486, 6.84592805818111305154925657091, 7.16439233009744494643733664065, 8.626488613705784786925293268852, 9.116540003014000453734502108780, 10.35041658068867605784508721036, 11.27901296185039558727708681889

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