Properties

Label 2-177-1.1-c11-0-52
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

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3.32·2-s − 243·3-s − 2.03e3·4-s + 1.03e4·5-s + 807.·6-s + 3.60e4·7-s + 1.35e4·8-s + 5.90e4·9-s − 3.45e4·10-s + 2.10e5·11-s + 4.94e5·12-s − 1.65e5·13-s − 1.19e5·14-s − 2.52e6·15-s + 4.12e6·16-s + 1.10e7·17-s − 1.96e5·18-s + 6.36e6·19-s − 2.11e7·20-s − 8.74e6·21-s − 6.99e5·22-s + 3.65e7·23-s − 3.29e6·24-s + 5.90e7·25-s + 5.49e5·26-s − 1.43e7·27-s − 7.33e7·28-s + ⋯
L(s)  = 1  − 0.0734·2-s − 0.577·3-s − 0.994·4-s + 1.48·5-s + 0.0423·6-s + 0.809·7-s + 0.146·8-s + 0.333·9-s − 0.109·10-s + 0.394·11-s + 0.574·12-s − 0.123·13-s − 0.0594·14-s − 0.858·15-s + 0.983·16-s + 1.89·17-s − 0.0244·18-s + 0.589·19-s − 1.47·20-s − 0.467·21-s − 0.0289·22-s + 1.18·23-s − 0.0845·24-s + 1.21·25-s + 0.00906·26-s − 0.192·27-s − 0.805·28-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

\(L(6)\) \(\approx\) \(2.720714540\)
\(L(\frac12)\) \(\approx\) \(2.720714540\)
\(L(\frac{13}{2})\) 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 \( 1 + 3.32T + 2.04e3T^{2} \)
5 \( 1 - 1.03e4T + 4.88e7T^{2} \)
7 \( 1 - 3.60e4T + 1.97e9T^{2} \)
11 \( 1 - 2.10e5T + 2.85e11T^{2} \)
13 \( 1 + 1.65e5T + 1.79e12T^{2} \)
17 \( 1 - 1.10e7T + 3.42e13T^{2} \)
19 \( 1 - 6.36e6T + 1.16e14T^{2} \)
23 \( 1 - 3.65e7T + 9.52e14T^{2} \)
29 \( 1 - 4.69e7T + 1.22e16T^{2} \)
31 \( 1 - 2.92e8T + 2.54e16T^{2} \)
37 \( 1 + 8.53e7T + 1.77e17T^{2} \)
41 \( 1 + 1.98e8T + 5.50e17T^{2} \)
43 \( 1 + 1.07e9T + 9.29e17T^{2} \)
47 \( 1 - 4.33e8T + 2.47e18T^{2} \)
53 \( 1 - 5.70e7T + 9.26e18T^{2} \)
61 \( 1 + 7.10e9T + 4.35e19T^{2} \)
67 \( 1 + 1.47e10T + 1.22e20T^{2} \)
71 \( 1 + 4.78e8T + 2.31e20T^{2} \)
73 \( 1 - 1.94e10T + 3.13e20T^{2} \)
79 \( 1 + 1.66e10T + 7.47e20T^{2} \)
83 \( 1 - 1.27e10T + 1.28e21T^{2} \)
89 \( 1 - 2.31e10T + 2.77e21T^{2} \)
97 \( 1 + 7.93e10T + 7.15e21T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.29625094205070099711785964814, −9.842533350674839084416506554871, −8.843997271989220040786685929988, −7.70710396216186325528217113277, −6.27357426273313093675091103583, −5.31819195831311908461031894211, −4.76540239250023413185274407857, −3.14176556838132930471202946149, −1.47609094327459646299838759636, −0.916898562946390714450685344364, 0.916898562946390714450685344364, 1.47609094327459646299838759636, 3.14176556838132930471202946149, 4.76540239250023413185274407857, 5.31819195831311908461031894211, 6.27357426273313093675091103583, 7.70710396216186325528217113277, 8.843997271989220040786685929988, 9.842533350674839084416506554871, 10.29625094205070099711785964814

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