Properties

Label 2-177-1.1-c9-0-18
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  − 6.00·2-s + 81·3-s − 475.·4-s + 2.08e3·5-s − 486.·6-s − 6.97e3·7-s + 5.93e3·8-s + 6.56e3·9-s − 1.25e4·10-s − 3.93e4·11-s − 3.85e4·12-s − 1.85e5·13-s + 4.18e4·14-s + 1.68e5·15-s + 2.08e5·16-s + 3.11e5·17-s − 3.94e4·18-s − 3.20e5·19-s − 9.92e5·20-s − 5.64e5·21-s + 2.36e5·22-s + 5.73e5·23-s + 4.80e5·24-s + 2.39e6·25-s + 1.11e6·26-s + 5.31e5·27-s + 3.31e6·28-s + ⋯
L(s)  = 1  − 0.265·2-s + 0.577·3-s − 0.929·4-s + 1.49·5-s − 0.153·6-s − 1.09·7-s + 0.512·8-s + 0.333·9-s − 0.396·10-s − 0.810·11-s − 0.536·12-s − 1.80·13-s + 0.291·14-s + 0.861·15-s + 0.793·16-s + 0.905·17-s − 0.0884·18-s − 0.563·19-s − 1.38·20-s − 0.633·21-s + 0.215·22-s + 0.427·23-s + 0.295·24-s + 1.22·25-s + 0.478·26-s + 0.192·27-s + 1.01·28-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

\(L(5)\) \(\approx\) \(1.711514522\)
\(L(\frac12)\) \(\approx\) \(1.711514522\)
\(L(\frac{11}{2})\) 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 \( 1 + 6.00T + 512T^{2} \)
5 \( 1 - 2.08e3T + 1.95e6T^{2} \)
7 \( 1 + 6.97e3T + 4.03e7T^{2} \)
11 \( 1 + 3.93e4T + 2.35e9T^{2} \)
13 \( 1 + 1.85e5T + 1.06e10T^{2} \)
17 \( 1 - 3.11e5T + 1.18e11T^{2} \)
19 \( 1 + 3.20e5T + 3.22e11T^{2} \)
23 \( 1 - 5.73e5T + 1.80e12T^{2} \)
29 \( 1 - 8.96e5T + 1.45e13T^{2} \)
31 \( 1 - 7.63e6T + 2.64e13T^{2} \)
37 \( 1 + 1.40e7T + 1.29e14T^{2} \)
41 \( 1 - 7.43e6T + 3.27e14T^{2} \)
43 \( 1 - 3.38e7T + 5.02e14T^{2} \)
47 \( 1 + 3.46e7T + 1.11e15T^{2} \)
53 \( 1 - 2.21e7T + 3.29e15T^{2} \)
61 \( 1 - 1.30e7T + 1.16e16T^{2} \)
67 \( 1 - 2.44e8T + 2.72e16T^{2} \)
71 \( 1 - 3.04e8T + 4.58e16T^{2} \)
73 \( 1 - 2.17e8T + 5.88e16T^{2} \)
79 \( 1 + 4.29e7T + 1.19e17T^{2} \)
83 \( 1 + 8.56e8T + 1.86e17T^{2} \)
89 \( 1 + 6.49e8T + 3.50e17T^{2} \)
97 \( 1 - 1.14e9T + 7.60e17T^{2} \)
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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

−10.27848861304625800008728607479, −9.895318093753366726387978829038, −9.318911001743699623560532820171, −8.157883785770693726824662339055, −6.95245479032625381974095133213, −5.65284469557726248516436909992, −4.73641610898981448730363044531, −3.09786096355344379427059896922, −2.18445757168668246667150616980, −0.63915120491504381943035831894, 0.63915120491504381943035831894, 2.18445757168668246667150616980, 3.09786096355344379427059896922, 4.73641610898981448730363044531, 5.65284469557726248516436909992, 6.95245479032625381974095133213, 8.157883785770693726824662339055, 9.318911001743699623560532820171, 9.895318093753366726387978829038, 10.27848861304625800008728607479

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