Properties

Label 2-177-177.5-c2-0-8
Degree $2$
Conductor $177$
Sign $0.313 - 0.949i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.792 + 2.35i)2-s + (−2.47 − 1.69i)3-s + (−1.72 − 1.30i)4-s + (−7.18 − 2.86i)5-s + (5.95 − 4.47i)6-s + (10.9 − 5.04i)7-s + (−3.77 + 2.55i)8-s + (3.23 + 8.39i)9-s + (12.4 − 14.6i)10-s + (12.6 + 3.51i)11-s + (2.03 + 6.16i)12-s + (−8.60 + 16.2i)13-s + (3.22 + 29.6i)14-s + (12.9 + 19.2i)15-s + (−5.34 − 19.2i)16-s + (−0.124 + 0.268i)17-s + ⋯
L(s)  = 1  + (−0.396 + 1.17i)2-s + (−0.824 − 0.565i)3-s + (−0.430 − 0.327i)4-s + (−1.43 − 0.572i)5-s + (0.992 − 0.745i)6-s + (1.55 − 0.720i)7-s + (−0.471 + 0.319i)8-s + (0.359 + 0.933i)9-s + (1.24 − 1.46i)10-s + (1.15 + 0.319i)11-s + (0.169 + 0.513i)12-s + (−0.662 + 1.24i)13-s + (0.230 + 2.11i)14-s + (0.861 + 1.28i)15-s + (−0.333 − 1.20i)16-s + (−0.00729 + 0.0157i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.313 - 0.949i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ 0.313 - 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.696677 + 0.503507i\)
\(L(\frac12)\) \(\approx\) \(0.696677 + 0.503507i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.47 + 1.69i)T \)
59 \( 1 + (50.5 + 30.4i)T \)
good2 \( 1 + (0.792 - 2.35i)T + (-3.18 - 2.42i)T^{2} \)
5 \( 1 + (7.18 + 2.86i)T + (18.1 + 17.1i)T^{2} \)
7 \( 1 + (-10.9 + 5.04i)T + (31.7 - 37.3i)T^{2} \)
11 \( 1 + (-12.6 - 3.51i)T + (103. + 62.3i)T^{2} \)
13 \( 1 + (8.60 - 16.2i)T + (-94.8 - 139. i)T^{2} \)
17 \( 1 + (0.124 - 0.268i)T + (-187. - 220. i)T^{2} \)
19 \( 1 + (-20.6 + 4.54i)T + (327. - 151. i)T^{2} \)
23 \( 1 + (-31.7 - 5.20i)T + (501. + 168. i)T^{2} \)
29 \( 1 + (-4.51 - 13.3i)T + (-669. + 508. i)T^{2} \)
31 \( 1 + (-8.57 - 1.88i)T + (872. + 403. i)T^{2} \)
37 \( 1 + (15.2 - 22.4i)T + (-506. - 1.27e3i)T^{2} \)
41 \( 1 + (-34.3 + 5.62i)T + (1.59e3 - 536. i)T^{2} \)
43 \( 1 + (-7.45 - 26.8i)T + (-1.58e3 + 953. i)T^{2} \)
47 \( 1 + (-40.8 + 16.2i)T + (1.60e3 - 1.51e3i)T^{2} \)
53 \( 1 + (10.2 - 8.70i)T + (454. - 2.77e3i)T^{2} \)
61 \( 1 + (-40.8 - 13.7i)T + (2.96e3 + 2.25e3i)T^{2} \)
67 \( 1 + (-9.22 - 13.6i)T + (-1.66e3 + 4.17e3i)T^{2} \)
71 \( 1 + (-80.8 + 32.1i)T + (3.65e3 - 3.46e3i)T^{2} \)
73 \( 1 + (40.8 - 4.44i)T + (5.20e3 - 1.14e3i)T^{2} \)
79 \( 1 + (18.4 - 11.0i)T + (2.92e3 - 5.51e3i)T^{2} \)
83 \( 1 + (53.3 - 2.89i)T + (6.84e3 - 744. i)T^{2} \)
89 \( 1 + (18.4 + 54.6i)T + (-6.30e3 + 4.79e3i)T^{2} \)
97 \( 1 + (144. + 15.6i)T + (9.18e3 + 2.02e3i)T^{2} \)
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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

−12.25934353373264670475830773710, −11.64501984372660005413692961429, −11.17778599293356467342835772083, −9.179718307122340600342391881704, −8.138629848110319448546375268220, −7.31561916307098717655692558138, −6.90992738743776781749611804337, −5.09564128086325540636143432177, −4.38763002495645318014298372263, −1.13530011856353039668872448586, 0.886087769889432818297453037468, 3.04818249381057320191913136857, 4.24851930288292480305417903148, 5.55646305078139351606247459959, 7.19332806888354621639858348947, 8.412004461335357077954161101645, 9.515419879908684152632582229559, 10.87136532316848646588244095349, 11.13115329792552580707082935990, 11.98781954828776078562355063682

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