Properties

Label 2-177-1.1-c9-0-53
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16.6·2-s + 81·3-s − 233.·4-s − 1.23e3·5-s − 1.35e3·6-s + 5.16e3·7-s + 1.24e4·8-s + 6.56e3·9-s + 2.06e4·10-s + 3.50e4·11-s − 1.89e4·12-s − 1.50e5·13-s − 8.61e4·14-s − 1.00e5·15-s − 8.81e4·16-s − 7.78e4·17-s − 1.09e5·18-s − 9.75e5·19-s + 2.89e5·20-s + 4.18e5·21-s − 5.84e5·22-s + 1.25e6·23-s + 1.00e6·24-s − 4.19e5·25-s + 2.50e6·26-s + 5.31e5·27-s − 1.20e6·28-s + ⋯
L(s)  = 1  − 0.737·2-s + 0.577·3-s − 0.455·4-s − 0.886·5-s − 0.425·6-s + 0.812·7-s + 1.07·8-s + 0.333·9-s + 0.653·10-s + 0.721·11-s − 0.263·12-s − 1.45·13-s − 0.599·14-s − 0.511·15-s − 0.336·16-s − 0.225·17-s − 0.245·18-s − 1.71·19-s + 0.404·20-s + 0.469·21-s − 0.532·22-s + 0.935·23-s + 0.620·24-s − 0.214·25-s + 1.07·26-s + 0.192·27-s − 0.370·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 + 16.6T + 512T^{2} \)
5 \( 1 + 1.23e3T + 1.95e6T^{2} \)
7 \( 1 - 5.16e3T + 4.03e7T^{2} \)
11 \( 1 - 3.50e4T + 2.35e9T^{2} \)
13 \( 1 + 1.50e5T + 1.06e10T^{2} \)
17 \( 1 + 7.78e4T + 1.18e11T^{2} \)
19 \( 1 + 9.75e5T + 3.22e11T^{2} \)
23 \( 1 - 1.25e6T + 1.80e12T^{2} \)
29 \( 1 - 4.34e6T + 1.45e13T^{2} \)
31 \( 1 - 8.89e6T + 2.64e13T^{2} \)
37 \( 1 - 1.60e7T + 1.29e14T^{2} \)
41 \( 1 - 4.19e6T + 3.27e14T^{2} \)
43 \( 1 - 1.36e7T + 5.02e14T^{2} \)
47 \( 1 + 4.18e7T + 1.11e15T^{2} \)
53 \( 1 - 1.42e7T + 3.29e15T^{2} \)
61 \( 1 - 1.20e7T + 1.16e16T^{2} \)
67 \( 1 - 1.80e8T + 2.72e16T^{2} \)
71 \( 1 + 4.01e8T + 4.58e16T^{2} \)
73 \( 1 + 3.63e8T + 5.88e16T^{2} \)
79 \( 1 - 3.93e8T + 1.19e17T^{2} \)
83 \( 1 + 3.21e8T + 1.86e17T^{2} \)
89 \( 1 - 9.37e8T + 3.50e17T^{2} \)
97 \( 1 + 1.46e9T + 7.60e17T^{2} \)
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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

−10.31148702422114340525022883394, −9.349535553824140778771822012094, −8.367185252850313118095232340770, −7.88408694019301035556686665489, −6.76728581938700881556670907480, −4.67316538916193748122426315722, −4.25642777555833228709851930093, −2.53987752792308548554984156946, −1.18367195454656937757153291683, 0, 1.18367195454656937757153291683, 2.53987752792308548554984156946, 4.25642777555833228709851930093, 4.67316538916193748122426315722, 6.76728581938700881556670907480, 7.88408694019301035556686665489, 8.367185252850313118095232340770, 9.349535553824140778771822012094, 10.31148702422114340525022883394

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