Properties

Label 2-177-1.1-c5-0-31
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.75·2-s + 9·3-s − 24.4·4-s + 5.39·5-s − 24.8·6-s − 153.·7-s + 155.·8-s + 81·9-s − 14.8·10-s + 761.·11-s − 219.·12-s + 217.·13-s + 423.·14-s + 48.5·15-s + 352.·16-s − 1.05e3·17-s − 223.·18-s − 2.36e3·19-s − 131.·20-s − 1.38e3·21-s − 2.09e3·22-s + 3.57e3·23-s + 1.39e3·24-s − 3.09e3·25-s − 600.·26-s + 729·27-s + 3.74e3·28-s + ⋯
L(s)  = 1  − 0.487·2-s + 0.577·3-s − 0.762·4-s + 0.0965·5-s − 0.281·6-s − 1.18·7-s + 0.858·8-s + 0.333·9-s − 0.0470·10-s + 1.89·11-s − 0.440·12-s + 0.357·13-s + 0.576·14-s + 0.0557·15-s + 0.343·16-s − 0.888·17-s − 0.162·18-s − 1.50·19-s − 0.0735·20-s − 0.683·21-s − 0.924·22-s + 1.40·23-s + 0.495·24-s − 0.990·25-s − 0.174·26-s + 0.192·27-s + 0.902·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
59 \( 1 - 3.48e3T \)
good2 \( 1 + 2.75T + 32T^{2} \)
5 \( 1 - 5.39T + 3.12e3T^{2} \)
7 \( 1 + 153.T + 1.68e4T^{2} \)
11 \( 1 - 761.T + 1.61e5T^{2} \)
13 \( 1 - 217.T + 3.71e5T^{2} \)
17 \( 1 + 1.05e3T + 1.41e6T^{2} \)
19 \( 1 + 2.36e3T + 2.47e6T^{2} \)
23 \( 1 - 3.57e3T + 6.43e6T^{2} \)
29 \( 1 - 538.T + 2.05e7T^{2} \)
31 \( 1 + 6.08e3T + 2.86e7T^{2} \)
37 \( 1 + 1.26e4T + 6.93e7T^{2} \)
41 \( 1 + 3.78e3T + 1.15e8T^{2} \)
43 \( 1 - 2.07e4T + 1.47e8T^{2} \)
47 \( 1 + 1.97e4T + 2.29e8T^{2} \)
53 \( 1 + 1.80e4T + 4.18e8T^{2} \)
61 \( 1 + 8.69e3T + 8.44e8T^{2} \)
67 \( 1 + 2.66e4T + 1.35e9T^{2} \)
71 \( 1 + 1.66e4T + 1.80e9T^{2} \)
73 \( 1 + 5.79e4T + 2.07e9T^{2} \)
79 \( 1 + 3.47e4T + 3.07e9T^{2} \)
83 \( 1 + 5.80e4T + 3.93e9T^{2} \)
89 \( 1 + 2.77e4T + 5.58e9T^{2} \)
97 \( 1 - 6.47e4T + 8.58e9T^{2} \)
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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

−11.10196019764883245676460308852, −9.960604791953435523353342308357, −8.988817482778570097664652929885, −8.819437871086016907880173796766, −7.13430540505529702871086143761, −6.22372403410979758130240271794, −4.36161309107508631978953191242, −3.49060940722434141366489623569, −1.59982124300523980286507738238, 0, 1.59982124300523980286507738238, 3.49060940722434141366489623569, 4.36161309107508631978953191242, 6.22372403410979758130240271794, 7.13430540505529702871086143761, 8.819437871086016907880173796766, 8.988817482778570097664652929885, 9.960604791953435523353342308357, 11.10196019764883245676460308852

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