# Properties

 Label 2-177-1.1-c9-0-79 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 $1$

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## Dirichlet series

 L(s)  = 1 + 41.8·2-s − 81·3-s + 1.24e3·4-s + 372.·5-s − 3.39e3·6-s − 8.19e3·7-s + 3.05e4·8-s + 6.56e3·9-s + 1.55e4·10-s − 8.23e4·11-s − 1.00e5·12-s + 2.03e5·13-s − 3.43e5·14-s − 3.01e4·15-s + 6.43e5·16-s − 2.24e5·17-s + 2.74e5·18-s − 2.25e5·19-s + 4.62e5·20-s + 6.64e5·21-s − 3.44e6·22-s − 1.73e6·23-s − 2.47e6·24-s − 1.81e6·25-s + 8.52e6·26-s − 5.31e5·27-s − 1.01e7·28-s + ⋯
 L(s)  = 1 + 1.85·2-s − 0.577·3-s + 2.42·4-s + 0.266·5-s − 1.06·6-s − 1.29·7-s + 2.63·8-s + 0.333·9-s + 0.493·10-s − 1.69·11-s − 1.39·12-s + 1.97·13-s − 2.38·14-s − 0.153·15-s + 2.45·16-s − 0.651·17-s + 0.616·18-s − 0.396·19-s + 0.646·20-s + 0.745·21-s − 3.13·22-s − 1.28·23-s − 1.52·24-s − 0.928·25-s + 3.65·26-s − 0.192·27-s − 3.12·28-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 - 41.8T + 512T^{2}$$
5 $$1 - 372.T + 1.95e6T^{2}$$
7 $$1 + 8.19e3T + 4.03e7T^{2}$$
11 $$1 + 8.23e4T + 2.35e9T^{2}$$
13 $$1 - 2.03e5T + 1.06e10T^{2}$$
17 $$1 + 2.24e5T + 1.18e11T^{2}$$
19 $$1 + 2.25e5T + 3.22e11T^{2}$$
23 $$1 + 1.73e6T + 1.80e12T^{2}$$
29 $$1 - 9.98e5T + 1.45e13T^{2}$$
31 $$1 - 3.18e6T + 2.64e13T^{2}$$
37 $$1 + 1.44e7T + 1.29e14T^{2}$$
41 $$1 - 1.46e7T + 3.27e14T^{2}$$
43 $$1 + 4.45e7T + 5.02e14T^{2}$$
47 $$1 + 3.24e7T + 1.11e15T^{2}$$
53 $$1 + 1.75e7T + 3.29e15T^{2}$$
61 $$1 - 4.48e7T + 1.16e16T^{2}$$
67 $$1 - 1.16e8T + 2.72e16T^{2}$$
71 $$1 - 2.10e8T + 4.58e16T^{2}$$
73 $$1 + 2.37e8T + 5.88e16T^{2}$$
79 $$1 + 2.98e8T + 1.19e17T^{2}$$
83 $$1 - 2.78e8T + 1.86e17T^{2}$$
89 $$1 - 6.35e8T + 3.50e17T^{2}$$
97 $$1 + 1.37e9T + 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

−10.79971850144204604587957229300, −10.07431382531293151298438696808, −8.158193970659170403883534633596, −6.62898505000689170700321513408, −6.12996562181438395144941895898, −5.28830978047468504475037554670, −4.01757755332346036564560110655, −3.12563032345391349440706965887, −1.91558550631045613031247957128, 0, 1.91558550631045613031247957128, 3.12563032345391349440706965887, 4.01757755332346036564560110655, 5.28830978047468504475037554670, 6.12996562181438395144941895898, 6.62898505000689170700321513408, 8.158193970659170403883534633596, 10.07431382531293151298438696808, 10.79971850144204604587957229300