Properties

Label 2-177-1.1-c11-0-100
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 77.8·2-s − 243·3-s + 4.01e3·4-s + 5.39e3·5-s − 1.89e4·6-s − 4.28e4·7-s + 1.53e5·8-s + 5.90e4·9-s + 4.20e5·10-s − 1.04e5·11-s − 9.75e5·12-s − 1.47e6·13-s − 3.33e6·14-s − 1.31e6·15-s + 3.71e6·16-s + 8.81e6·17-s + 4.59e6·18-s + 6.43e6·19-s + 2.16e7·20-s + 1.04e7·21-s − 8.14e6·22-s − 4.37e7·23-s − 3.72e7·24-s − 1.97e7·25-s − 1.14e8·26-s − 1.43e7·27-s − 1.71e8·28-s + ⋯
L(s)  = 1  + 1.72·2-s − 0.577·3-s + 1.96·4-s + 0.772·5-s − 0.993·6-s − 0.962·7-s + 1.65·8-s + 0.333·9-s + 1.32·10-s − 0.195·11-s − 1.13·12-s − 1.09·13-s − 1.65·14-s − 0.445·15-s + 0.884·16-s + 1.50·17-s + 0.573·18-s + 0.596·19-s + 1.51·20-s + 0.555·21-s − 0.337·22-s − 1.41·23-s − 0.954·24-s − 0.403·25-s − 1.89·26-s − 0.192·27-s − 1.88·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
59 \( 1 - 7.14e8T \)
good2 \( 1 - 77.8T + 2.04e3T^{2} \)
5 \( 1 - 5.39e3T + 4.88e7T^{2} \)
7 \( 1 + 4.28e4T + 1.97e9T^{2} \)
11 \( 1 + 1.04e5T + 2.85e11T^{2} \)
13 \( 1 + 1.47e6T + 1.79e12T^{2} \)
17 \( 1 - 8.81e6T + 3.42e13T^{2} \)
19 \( 1 - 6.43e6T + 1.16e14T^{2} \)
23 \( 1 + 4.37e7T + 9.52e14T^{2} \)
29 \( 1 - 5.52e7T + 1.22e16T^{2} \)
31 \( 1 + 1.63e8T + 2.54e16T^{2} \)
37 \( 1 + 3.81e8T + 1.77e17T^{2} \)
41 \( 1 - 1.35e8T + 5.50e17T^{2} \)
43 \( 1 + 3.37e8T + 9.29e17T^{2} \)
47 \( 1 + 5.89e8T + 2.47e18T^{2} \)
53 \( 1 + 2.20e9T + 9.26e18T^{2} \)
61 \( 1 - 6.83e8T + 4.35e19T^{2} \)
67 \( 1 - 1.58e10T + 1.22e20T^{2} \)
71 \( 1 + 8.59e9T + 2.31e20T^{2} \)
73 \( 1 - 2.85e9T + 3.13e20T^{2} \)
79 \( 1 + 2.26e10T + 7.47e20T^{2} \)
83 \( 1 + 4.45e10T + 1.28e21T^{2} \)
89 \( 1 + 2.58e10T + 2.77e21T^{2} \)
97 \( 1 - 6.65e10T + 7.15e21T^{2} \)
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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

−10.21930658105727865752987285519, −9.682864017160521852787101245015, −7.58558515111424858054046248679, −6.56707572615739794729983040058, −5.71028239542102278719270305111, −5.15438719399183140310602635364, −3.82088032359173884376059668508, −2.86460326360819418600341123272, −1.73176009269148747249902002999, 0, 1.73176009269148747249902002999, 2.86460326360819418600341123272, 3.82088032359173884376059668508, 5.15438719399183140310602635364, 5.71028239542102278719270305111, 6.56707572615739794729983040058, 7.58558515111424858054046248679, 9.682864017160521852787101245015, 10.21930658105727865752987285519

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