Properties

Label 2-177-1.1-c11-0-76
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 $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 20.9·2-s − 243·3-s − 1.60e3·4-s + 5.79e3·5-s − 5.09e3·6-s + 2.21e4·7-s − 7.67e4·8-s + 5.90e4·9-s + 1.21e5·10-s + 9.36e5·11-s + 3.90e5·12-s − 1.11e6·13-s + 4.65e5·14-s − 1.40e6·15-s + 1.68e6·16-s − 9.92e6·17-s + 1.23e6·18-s − 1.60e7·19-s − 9.32e6·20-s − 5.38e6·21-s + 1.96e7·22-s − 9.19e6·23-s + 1.86e7·24-s − 1.51e7·25-s − 2.33e7·26-s − 1.43e7·27-s − 3.56e7·28-s + ⋯
L(s)  = 1  + 0.463·2-s − 0.577·3-s − 0.785·4-s + 0.829·5-s − 0.267·6-s + 0.498·7-s − 0.827·8-s + 0.333·9-s + 0.384·10-s + 1.75·11-s + 0.453·12-s − 0.832·13-s + 0.231·14-s − 0.479·15-s + 0.401·16-s − 1.69·17-s + 0.154·18-s − 1.49·19-s − 0.651·20-s − 0.287·21-s + 0.812·22-s − 0.298·23-s + 0.477·24-s − 0.311·25-s − 0.385·26-s − 0.192·27-s − 0.391·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 - 20.9T + 2.04e3T^{2} \)
5 \( 1 - 5.79e3T + 4.88e7T^{2} \)
7 \( 1 - 2.21e4T + 1.97e9T^{2} \)
11 \( 1 - 9.36e5T + 2.85e11T^{2} \)
13 \( 1 + 1.11e6T + 1.79e12T^{2} \)
17 \( 1 + 9.92e6T + 3.42e13T^{2} \)
19 \( 1 + 1.60e7T + 1.16e14T^{2} \)
23 \( 1 + 9.19e6T + 9.52e14T^{2} \)
29 \( 1 - 1.93e8T + 1.22e16T^{2} \)
31 \( 1 - 2.28e8T + 2.54e16T^{2} \)
37 \( 1 - 6.39e8T + 1.77e17T^{2} \)
41 \( 1 - 2.59e8T + 5.50e17T^{2} \)
43 \( 1 + 5.30e6T + 9.29e17T^{2} \)
47 \( 1 - 9.16e8T + 2.47e18T^{2} \)
53 \( 1 + 4.20e9T + 9.26e18T^{2} \)
61 \( 1 + 8.56e9T + 4.35e19T^{2} \)
67 \( 1 - 1.23e10T + 1.22e20T^{2} \)
71 \( 1 + 4.22e9T + 2.31e20T^{2} \)
73 \( 1 + 6.29e9T + 3.13e20T^{2} \)
79 \( 1 + 5.22e10T + 7.47e20T^{2} \)
83 \( 1 + 7.62e9T + 1.28e21T^{2} \)
89 \( 1 + 9.11e10T + 2.77e21T^{2} \)
97 \( 1 - 1.48e11T + 7.15e21T^{2} \)
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

−10.06803755423998834034997516603, −9.253293045316136502814325475398, −8.362984368550223201619578924627, −6.54265276920972073378508737513, −6.11947937124597838906675477407, −4.53786911177669483302783797092, −4.38186852947030872884860017969, −2.46886523291150976120521769819, −1.25463636889220777619617458450, 0, 1.25463636889220777619617458450, 2.46886523291150976120521769819, 4.38186852947030872884860017969, 4.53786911177669483302783797092, 6.11947937124597838906675477407, 6.54265276920972073378508737513, 8.362984368550223201619578924627, 9.253293045316136502814325475398, 10.06803755423998834034997516603

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