Properties

Label 2-177-1.1-c11-0-27
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 67.4·2-s + 243·3-s + 2.50e3·4-s − 169.·5-s − 1.63e4·6-s + 6.57e4·7-s − 3.07e4·8-s + 5.90e4·9-s + 1.14e4·10-s − 8.51e5·11-s + 6.08e5·12-s − 6.45e4·13-s − 4.43e6·14-s − 4.11e4·15-s − 3.05e6·16-s + 4.48e6·17-s − 3.98e6·18-s − 8.64e6·19-s − 4.24e5·20-s + 1.59e7·21-s + 5.74e7·22-s − 8.74e6·23-s − 7.47e6·24-s − 4.87e7·25-s + 4.35e6·26-s + 1.43e7·27-s + 1.64e8·28-s + ⋯
L(s)  = 1  − 1.49·2-s + 0.577·3-s + 1.22·4-s − 0.0242·5-s − 0.860·6-s + 1.47·7-s − 0.331·8-s + 0.333·9-s + 0.0361·10-s − 1.59·11-s + 0.705·12-s − 0.0482·13-s − 2.20·14-s − 0.0140·15-s − 0.727·16-s + 0.765·17-s − 0.496·18-s − 0.800·19-s − 0.0296·20-s + 0.853·21-s + 2.37·22-s − 0.283·23-s − 0.191·24-s − 0.999·25-s + 0.0719·26-s + 0.192·27-s + 1.80·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(1.257117332\)
\(L(\frac12)\) \(\approx\) \(1.257117332\)
\(L(\frac{13}{2})\) 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 \( 1 + 67.4T + 2.04e3T^{2} \)
5 \( 1 + 169.T + 4.88e7T^{2} \)
7 \( 1 - 6.57e4T + 1.97e9T^{2} \)
11 \( 1 + 8.51e5T + 2.85e11T^{2} \)
13 \( 1 + 6.45e4T + 1.79e12T^{2} \)
17 \( 1 - 4.48e6T + 3.42e13T^{2} \)
19 \( 1 + 8.64e6T + 1.16e14T^{2} \)
23 \( 1 + 8.74e6T + 9.52e14T^{2} \)
29 \( 1 - 1.62e8T + 1.22e16T^{2} \)
31 \( 1 + 2.58e8T + 2.54e16T^{2} \)
37 \( 1 - 6.12e7T + 1.77e17T^{2} \)
41 \( 1 - 1.09e9T + 5.50e17T^{2} \)
43 \( 1 - 2.73e8T + 9.29e17T^{2} \)
47 \( 1 - 2.85e8T + 2.47e18T^{2} \)
53 \( 1 - 3.57e9T + 9.26e18T^{2} \)
61 \( 1 + 5.00e9T + 4.35e19T^{2} \)
67 \( 1 - 1.14e10T + 1.22e20T^{2} \)
71 \( 1 - 1.70e10T + 2.31e20T^{2} \)
73 \( 1 + 1.15e9T + 3.13e20T^{2} \)
79 \( 1 - 5.14e8T + 7.47e20T^{2} \)
83 \( 1 + 4.94e9T + 1.28e21T^{2} \)
89 \( 1 + 6.81e10T + 2.77e21T^{2} \)
97 \( 1 + 7.84e10T + 7.15e21T^{2} \)
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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

−10.53064988556413975958746435434, −9.590242923332630636005324979287, −8.396001599482433578473127872623, −8.010226445193033399578515840315, −7.28186792541692363039317745729, −5.50463479604220806666768657454, −4.31596398969455363727299753429, −2.52614358278414961161624312084, −1.77302296662823510489516392548, −0.62352861414663876489159609705, 0.62352861414663876489159609705, 1.77302296662823510489516392548, 2.52614358278414961161624312084, 4.31596398969455363727299753429, 5.50463479604220806666768657454, 7.28186792541692363039317745729, 8.010226445193033399578515840315, 8.396001599482433578473127872623, 9.590242923332630636005324979287, 10.53064988556413975958746435434

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