Properties

Label 2-177-1.1-c9-0-44
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  − 20.0·2-s + 81·3-s − 109.·4-s + 2.61e3·5-s − 1.62e3·6-s − 4.64e3·7-s + 1.24e4·8-s + 6.56e3·9-s − 5.25e4·10-s + 9.43e4·11-s − 8.86e3·12-s + 1.01e5·13-s + 9.31e4·14-s + 2.12e5·15-s − 1.94e5·16-s − 1.74e5·17-s − 1.31e5·18-s + 5.98e5·19-s − 2.86e5·20-s − 3.76e5·21-s − 1.89e6·22-s − 8.11e5·23-s + 1.00e6·24-s + 4.91e6·25-s − 2.04e6·26-s + 5.31e5·27-s + 5.08e5·28-s + ⋯
L(s)  = 1  − 0.886·2-s + 0.577·3-s − 0.213·4-s + 1.87·5-s − 0.511·6-s − 0.730·7-s + 1.07·8-s + 0.333·9-s − 1.66·10-s + 1.94·11-s − 0.123·12-s + 0.989·13-s + 0.648·14-s + 1.08·15-s − 0.740·16-s − 0.507·17-s − 0.295·18-s + 1.05·19-s − 0.400·20-s − 0.421·21-s − 1.72·22-s − 0.604·23-s + 0.621·24-s + 2.51·25-s − 0.877·26-s + 0.192·27-s + 0.156·28-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

\(L(5)\) \(\approx\) \(2.766000201\)
\(L(\frac12)\) \(\approx\) \(2.766000201\)
\(L(\frac{11}{2})\) 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 \( 1 + 20.0T + 512T^{2} \)
5 \( 1 - 2.61e3T + 1.95e6T^{2} \)
7 \( 1 + 4.64e3T + 4.03e7T^{2} \)
11 \( 1 - 9.43e4T + 2.35e9T^{2} \)
13 \( 1 - 1.01e5T + 1.06e10T^{2} \)
17 \( 1 + 1.74e5T + 1.18e11T^{2} \)
19 \( 1 - 5.98e5T + 3.22e11T^{2} \)
23 \( 1 + 8.11e5T + 1.80e12T^{2} \)
29 \( 1 - 7.22e6T + 1.45e13T^{2} \)
31 \( 1 - 7.32e6T + 2.64e13T^{2} \)
37 \( 1 - 7.15e5T + 1.29e14T^{2} \)
41 \( 1 + 2.55e7T + 3.27e14T^{2} \)
43 \( 1 + 8.78e6T + 5.02e14T^{2} \)
47 \( 1 - 7.96e6T + 1.11e15T^{2} \)
53 \( 1 + 1.41e7T + 3.29e15T^{2} \)
61 \( 1 + 1.20e8T + 1.16e16T^{2} \)
67 \( 1 + 1.23e8T + 2.72e16T^{2} \)
71 \( 1 + 3.13e8T + 4.58e16T^{2} \)
73 \( 1 + 1.28e8T + 5.88e16T^{2} \)
79 \( 1 + 3.90e8T + 1.19e17T^{2} \)
83 \( 1 - 5.79e8T + 1.86e17T^{2} \)
89 \( 1 - 1.25e8T + 3.50e17T^{2} \)
97 \( 1 - 5.54e8T + 7.60e17T^{2} \)
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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

−10.37344678001307162271910841157, −9.776843954184321264540799450655, −9.119502639830290409694085367090, −8.506015317642667712223048195956, −6.74808759502008402331162002247, −6.16468495600989213396388684533, −4.54536258494575455780272666856, −3.12051434965834285950230317446, −1.62571008837864230768823275097, −1.06113090025751423959505778816, 1.06113090025751423959505778816, 1.62571008837864230768823275097, 3.12051434965834285950230317446, 4.54536258494575455780272666856, 6.16468495600989213396388684533, 6.74808759502008402331162002247, 8.506015317642667712223048195956, 9.119502639830290409694085367090, 9.776843954184321264540799450655, 10.37344678001307162271910841157

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