Properties

Label 2-177-1.1-c9-0-54
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 $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 11.8·2-s − 81·3-s − 370.·4-s − 236.·5-s − 963.·6-s + 4.18e3·7-s − 1.05e4·8-s + 6.56e3·9-s − 2.81e3·10-s − 2.58e4·11-s + 3.00e4·12-s + 1.09e4·13-s + 4.98e4·14-s + 1.91e4·15-s + 6.47e4·16-s − 7.48e4·17-s + 7.80e4·18-s + 5.75e5·19-s + 8.77e4·20-s − 3.39e5·21-s − 3.07e5·22-s + 1.18e6·23-s + 8.50e5·24-s − 1.89e6·25-s + 1.30e5·26-s − 5.31e5·27-s − 1.55e6·28-s + ⋯
L(s)  = 1  + 0.525·2-s − 0.577·3-s − 0.723·4-s − 0.169·5-s − 0.303·6-s + 0.658·7-s − 0.906·8-s + 0.333·9-s − 0.0890·10-s − 0.531·11-s + 0.417·12-s + 0.106·13-s + 0.346·14-s + 0.0978·15-s + 0.246·16-s − 0.217·17-s + 0.175·18-s + 1.01·19-s + 0.122·20-s − 0.380·21-s − 0.279·22-s + 0.884·23-s + 0.523·24-s − 0.971·25-s + 0.0560·26-s − 0.192·27-s − 0.476·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - 11.8T + 512T^{2} \)
5 \( 1 + 236.T + 1.95e6T^{2} \)
7 \( 1 - 4.18e3T + 4.03e7T^{2} \)
11 \( 1 + 2.58e4T + 2.35e9T^{2} \)
13 \( 1 - 1.09e4T + 1.06e10T^{2} \)
17 \( 1 + 7.48e4T + 1.18e11T^{2} \)
19 \( 1 - 5.75e5T + 3.22e11T^{2} \)
23 \( 1 - 1.18e6T + 1.80e12T^{2} \)
29 \( 1 - 9.71e5T + 1.45e13T^{2} \)
31 \( 1 - 7.66e6T + 2.64e13T^{2} \)
37 \( 1 + 1.34e7T + 1.29e14T^{2} \)
41 \( 1 - 3.71e6T + 3.27e14T^{2} \)
43 \( 1 - 1.42e7T + 5.02e14T^{2} \)
47 \( 1 - 5.03e7T + 1.11e15T^{2} \)
53 \( 1 + 4.53e7T + 3.29e15T^{2} \)
61 \( 1 + 1.39e8T + 1.16e16T^{2} \)
67 \( 1 + 1.76e8T + 2.72e16T^{2} \)
71 \( 1 + 4.76e7T + 4.58e16T^{2} \)
73 \( 1 + 1.70e8T + 5.88e16T^{2} \)
79 \( 1 + 7.11e7T + 1.19e17T^{2} \)
83 \( 1 + 3.04e8T + 1.86e17T^{2} \)
89 \( 1 + 6.08e8T + 3.50e17T^{2} \)
97 \( 1 + 5.99e8T + 7.60e17T^{2} \)
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

−10.64372532793176872190527546431, −9.560542782628097239002629479264, −8.459360967377174663093272835514, −7.41231413876220956929696175901, −5.97540261002214412623807053004, −5.08345313880842115572227785118, −4.29192131271929068598317890488, −2.95911688963091854842478621133, −1.20294252736642915769046442990, 0, 1.20294252736642915769046442990, 2.95911688963091854842478621133, 4.29192131271929068598317890488, 5.08345313880842115572227785118, 5.97540261002214412623807053004, 7.41231413876220956929696175901, 8.459360967377174663093272835514, 9.560542782628097239002629479264, 10.64372532793176872190527546431

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