Properties

Label 2-177-1.1-c9-0-32
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 28.6·2-s + 81·3-s + 308.·4-s + 1.35e3·5-s − 2.32e3·6-s + 4.08e3·7-s + 5.82e3·8-s + 6.56e3·9-s − 3.87e4·10-s − 4.72e4·11-s + 2.49e4·12-s − 3.30e4·13-s − 1.16e5·14-s + 1.09e5·15-s − 3.24e5·16-s + 4.43e5·17-s − 1.87e5·18-s + 7.25e5·19-s + 4.17e5·20-s + 3.30e5·21-s + 1.35e6·22-s − 1.25e5·23-s + 4.72e5·24-s − 1.19e5·25-s + 9.46e5·26-s + 5.31e5·27-s + 1.26e6·28-s + ⋯
L(s)  = 1  − 1.26·2-s + 0.577·3-s + 0.602·4-s + 0.968·5-s − 0.730·6-s + 0.642·7-s + 0.503·8-s + 0.333·9-s − 1.22·10-s − 0.972·11-s + 0.347·12-s − 0.320·13-s − 0.813·14-s + 0.559·15-s − 1.23·16-s + 1.28·17-s − 0.421·18-s + 1.27·19-s + 0.583·20-s + 0.371·21-s + 1.23·22-s − 0.0933·23-s + 0.290·24-s − 0.0610·25-s + 0.406·26-s + 0.192·27-s + 0.387·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(1.856167768\)
\(L(\frac12)\) \(\approx\) \(1.856167768\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 + 28.6T + 512T^{2} \)
5 \( 1 - 1.35e3T + 1.95e6T^{2} \)
7 \( 1 - 4.08e3T + 4.03e7T^{2} \)
11 \( 1 + 4.72e4T + 2.35e9T^{2} \)
13 \( 1 + 3.30e4T + 1.06e10T^{2} \)
17 \( 1 - 4.43e5T + 1.18e11T^{2} \)
19 \( 1 - 7.25e5T + 3.22e11T^{2} \)
23 \( 1 + 1.25e5T + 1.80e12T^{2} \)
29 \( 1 + 4.03e6T + 1.45e13T^{2} \)
31 \( 1 - 8.12e6T + 2.64e13T^{2} \)
37 \( 1 - 1.61e7T + 1.29e14T^{2} \)
41 \( 1 - 2.56e7T + 3.27e14T^{2} \)
43 \( 1 + 5.08e6T + 5.02e14T^{2} \)
47 \( 1 - 3.17e6T + 1.11e15T^{2} \)
53 \( 1 - 2.86e6T + 3.29e15T^{2} \)
61 \( 1 + 1.34e8T + 1.16e16T^{2} \)
67 \( 1 + 2.02e8T + 2.72e16T^{2} \)
71 \( 1 + 2.73e8T + 4.58e16T^{2} \)
73 \( 1 + 4.58e7T + 5.88e16T^{2} \)
79 \( 1 - 1.56e7T + 1.19e17T^{2} \)
83 \( 1 - 4.78e8T + 1.86e17T^{2} \)
89 \( 1 - 5.24e8T + 3.50e17T^{2} \)
97 \( 1 - 5.11e8T + 7.60e17T^{2} \)
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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

−10.50209910256047024652910670185, −9.815427760833906593672338449280, −9.193474302966623790745405205205, −7.84518690960851067964565432953, −7.63790345754593795830513107330, −5.84748509157644298558397852741, −4.70912171122934630467463662513, −2.86364978008771631600496713757, −1.78144248657532102082781902364, −0.834240184080355285391524871476, 0.834240184080355285391524871476, 1.78144248657532102082781902364, 2.86364978008771631600496713757, 4.70912171122934630467463662513, 5.84748509157644298558397852741, 7.63790345754593795830513107330, 7.84518690960851067964565432953, 9.193474302966623790745405205205, 9.815427760833906593672338449280, 10.50209910256047024652910670185

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