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Properties

Label 2-177-59.58-c4-0-8
Degree $2$
Conductor $177$
Sign $-0.951 - 0.308i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.31i·2-s + 5.19·3-s + 10.6·4-s − 38.6·5-s + 12.0i·6-s + 9.48·7-s + 61.7i·8-s + 27·9-s − 89.6i·10-s − 58.4i·11-s + 55.1·12-s + 290. i·13-s + 22.0i·14-s − 200.·15-s + 26.7·16-s − 467.·17-s + ⋯
L(s)  = 1  + 0.579i·2-s + 0.577·3-s + 0.663·4-s − 1.54·5-s + 0.334i·6-s + 0.193·7-s + 0.964i·8-s + 0.333·9-s − 0.896i·10-s − 0.483i·11-s + 0.383·12-s + 1.72i·13-s + 0.112i·14-s − 0.892·15-s + 0.104·16-s − 1.61·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.951 - 0.308i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ -0.951 - 0.308i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.226051441\)
\(L(\frac12)\) \(\approx\) \(1.226051441\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.19T \)
59 \( 1 + (-3.31e3 - 1.07e3i)T \)
good2 \( 1 - 2.31iT - 16T^{2} \)
5 \( 1 + 38.6T + 625T^{2} \)
7 \( 1 - 9.48T + 2.40e3T^{2} \)
11 \( 1 + 58.4iT - 1.46e4T^{2} \)
13 \( 1 - 290. iT - 2.85e4T^{2} \)
17 \( 1 + 467.T + 8.35e4T^{2} \)
19 \( 1 + 261.T + 1.30e5T^{2} \)
23 \( 1 - 190. iT - 2.79e5T^{2} \)
29 \( 1 + 407.T + 7.07e5T^{2} \)
31 \( 1 - 1.89e3iT - 9.23e5T^{2} \)
37 \( 1 + 306. iT - 1.87e6T^{2} \)
41 \( 1 + 486.T + 2.82e6T^{2} \)
43 \( 1 + 1.83e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.57e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.71e3T + 7.89e6T^{2} \)
61 \( 1 - 3.95e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.26e3iT - 2.01e7T^{2} \)
71 \( 1 - 7.35e3T + 2.54e7T^{2} \)
73 \( 1 + 8.25e3iT - 2.83e7T^{2} \)
79 \( 1 - 8.38e3T + 3.89e7T^{2} \)
83 \( 1 + 8.21e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.36e4iT - 6.27e7T^{2} \)
97 \( 1 - 5.34e3iT - 8.85e7T^{2} \)
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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

−12.23879526762393826969249484009, −11.43743539232052382427287570354, −10.82264357692458777715863882807, −8.901321465658751806253164053513, −8.358950790259317451515806895184, −7.18783111408458813584462099827, −6.62479092535978975144336481490, −4.71049827281431532120802053314, −3.61364631698954893454446383922, −2.01265740823429642947135743960, 0.39000500222057168139181876293, 2.29312079033087916755719874759, 3.47238963151532784121071468532, 4.50519099405619102739880339377, 6.51038407182432642002179184467, 7.68113486305372587141701567579, 8.221489896747402903112409707386, 9.704053806703094777701693177437, 10.93427207750465555844376346868, 11.34504845327129548863548831292

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