Properties

Label 2-177-1.1-c11-0-56
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  − 40.3·2-s + 243·3-s − 415.·4-s + 1.36e4·5-s − 9.81e3·6-s + 8.10e4·7-s + 9.95e4·8-s + 5.90e4·9-s − 5.53e5·10-s + 2.43e5·11-s − 1.01e5·12-s − 2.52e6·13-s − 3.27e6·14-s + 3.32e6·15-s − 3.16e6·16-s − 6.01e6·17-s − 2.38e6·18-s + 1.38e7·19-s − 5.69e6·20-s + 1.97e7·21-s − 9.83e6·22-s + 3.48e7·23-s + 2.41e7·24-s + 1.38e8·25-s + 1.01e8·26-s + 1.43e7·27-s − 3.37e7·28-s + ⋯
L(s)  = 1  − 0.892·2-s + 0.577·3-s − 0.203·4-s + 1.96·5-s − 0.515·6-s + 1.82·7-s + 1.07·8-s + 0.333·9-s − 1.75·10-s + 0.455·11-s − 0.117·12-s − 1.88·13-s − 1.62·14-s + 1.13·15-s − 0.755·16-s − 1.02·17-s − 0.297·18-s + 1.28·19-s − 0.398·20-s + 1.05·21-s − 0.406·22-s + 1.12·23-s + 0.620·24-s + 2.84·25-s + 1.68·26-s + 0.192·27-s − 0.370·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

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

\(L(6)\) \(\approx\) \(3.255000048\)
\(L(\frac12)\) \(\approx\) \(3.255000048\)
\(L(\frac{13}{2})\) 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 \( 1 + 40.3T + 2.04e3T^{2} \)
5 \( 1 - 1.36e4T + 4.88e7T^{2} \)
7 \( 1 - 8.10e4T + 1.97e9T^{2} \)
11 \( 1 - 2.43e5T + 2.85e11T^{2} \)
13 \( 1 + 2.52e6T + 1.79e12T^{2} \)
17 \( 1 + 6.01e6T + 3.42e13T^{2} \)
19 \( 1 - 1.38e7T + 1.16e14T^{2} \)
23 \( 1 - 3.48e7T + 9.52e14T^{2} \)
29 \( 1 - 6.14e7T + 1.22e16T^{2} \)
31 \( 1 + 3.78e7T + 2.54e16T^{2} \)
37 \( 1 + 1.90e7T + 1.77e17T^{2} \)
41 \( 1 - 6.76e8T + 5.50e17T^{2} \)
43 \( 1 + 2.86e8T + 9.29e17T^{2} \)
47 \( 1 + 2.03e9T + 2.47e18T^{2} \)
53 \( 1 + 1.43e9T + 9.26e18T^{2} \)
61 \( 1 + 5.11e9T + 4.35e19T^{2} \)
67 \( 1 - 1.62e10T + 1.22e20T^{2} \)
71 \( 1 + 1.34e10T + 2.31e20T^{2} \)
73 \( 1 - 1.98e10T + 3.13e20T^{2} \)
79 \( 1 - 3.88e10T + 7.47e20T^{2} \)
83 \( 1 - 8.01e9T + 1.28e21T^{2} \)
89 \( 1 + 5.58e10T + 2.77e21T^{2} \)
97 \( 1 - 6.19e10T + 7.15e21T^{2} \)
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

−10.27435978907827799547021809756, −9.453815283868730881559070940014, −8.971432217796393756643550191394, −7.85072870214410109167166811720, −6.90457671491814486214505031997, −5.10433694394710662248395301366, −4.79759372101172297823241085047, −2.50638145172492918271485315619, −1.76481032591698096269380010868, −1.00724131998726806325373498528, 1.00724131998726806325373498528, 1.76481032591698096269380010868, 2.50638145172492918271485315619, 4.79759372101172297823241085047, 5.10433694394710662248395301366, 6.90457671491814486214505031997, 7.85072870214410109167166811720, 8.971432217796393756643550191394, 9.453815283868730881559070940014, 10.27435978907827799547021809756

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