Properties

Label 2-177-1.1-c3-0-1
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.254·2-s − 3·3-s − 7.93·4-s − 10.8·5-s − 0.763·6-s − 23.2·7-s − 4.05·8-s + 9·9-s − 2.75·10-s + 51.0·11-s + 23.8·12-s + 51.8·13-s − 5.92·14-s + 32.4·15-s + 62.4·16-s − 0.0421·17-s + 2.28·18-s − 85.2·19-s + 85.8·20-s + 69.8·21-s + 12.9·22-s − 8.50·23-s + 12.1·24-s − 7.87·25-s + 13.1·26-s − 27·27-s + 184.·28-s + ⋯
L(s)  = 1  + 0.0899·2-s − 0.577·3-s − 0.991·4-s − 0.967·5-s − 0.0519·6-s − 1.25·7-s − 0.179·8-s + 0.333·9-s − 0.0870·10-s + 1.39·11-s + 0.572·12-s + 1.10·13-s − 0.113·14-s + 0.558·15-s + 0.975·16-s − 0.000601·17-s + 0.0299·18-s − 1.02·19-s + 0.960·20-s + 0.726·21-s + 0.125·22-s − 0.0771·23-s + 0.103·24-s − 0.0629·25-s + 0.0995·26-s − 0.192·27-s + 1.24·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7238131545\)
\(L(\frac12)\) \(\approx\) \(0.7238131545\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
59 \( 1 + 59T \)
good2 \( 1 - 0.254T + 8T^{2} \)
5 \( 1 + 10.8T + 125T^{2} \)
7 \( 1 + 23.2T + 343T^{2} \)
11 \( 1 - 51.0T + 1.33e3T^{2} \)
13 \( 1 - 51.8T + 2.19e3T^{2} \)
17 \( 1 + 0.0421T + 4.91e3T^{2} \)
19 \( 1 + 85.2T + 6.85e3T^{2} \)
23 \( 1 + 8.50T + 1.21e4T^{2} \)
29 \( 1 + 101.T + 2.43e4T^{2} \)
31 \( 1 - 271.T + 2.97e4T^{2} \)
37 \( 1 + 9.54T + 5.06e4T^{2} \)
41 \( 1 + 185.T + 6.89e4T^{2} \)
43 \( 1 - 277.T + 7.95e4T^{2} \)
47 \( 1 - 309.T + 1.03e5T^{2} \)
53 \( 1 - 273.T + 1.48e5T^{2} \)
61 \( 1 - 752.T + 2.26e5T^{2} \)
67 \( 1 - 751.T + 3.00e5T^{2} \)
71 \( 1 + 651.T + 3.57e5T^{2} \)
73 \( 1 - 842.T + 3.89e5T^{2} \)
79 \( 1 + 368.T + 4.93e5T^{2} \)
83 \( 1 + 545.T + 5.71e5T^{2} \)
89 \( 1 - 634.T + 7.04e5T^{2} \)
97 \( 1 - 316.T + 9.12e5T^{2} \)
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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

−12.27668686885522155742338087688, −11.41940617268964662964489853589, −10.19130674825441353010004172234, −9.186649301197940382953391180470, −8.313429374314832524075528244344, −6.78098478081726529182643603989, −5.92270743709261570084638303101, −4.21746566999097889094382420512, −3.65393940995306566276635447984, −0.68288059924350781120312986834, 0.68288059924350781120312986834, 3.65393940995306566276635447984, 4.21746566999097889094382420512, 5.92270743709261570084638303101, 6.78098478081726529182643603989, 8.313429374314832524075528244344, 9.186649301197940382953391180470, 10.19130674825441353010004172234, 11.41940617268964662964489853589, 12.27668686885522155742338087688

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