Properties

Label 2-177-1.1-c9-0-41
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 $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 18.6·2-s − 81·3-s − 165.·4-s + 1.49e3·5-s − 1.50e3·6-s + 1.18e4·7-s − 1.26e4·8-s + 6.56e3·9-s + 2.77e4·10-s + 5.41e4·11-s + 1.34e4·12-s + 1.76e5·13-s + 2.19e5·14-s − 1.20e5·15-s − 1.49e5·16-s + 1.08e5·17-s + 1.22e5·18-s + 5.44e4·19-s − 2.47e5·20-s − 9.55e5·21-s + 1.00e6·22-s − 2.05e6·23-s + 1.02e6·24-s + 2.75e5·25-s + 3.27e6·26-s − 5.31e5·27-s − 1.95e6·28-s + ⋯
L(s)  = 1  + 0.822·2-s − 0.577·3-s − 0.323·4-s + 1.06·5-s − 0.474·6-s + 1.85·7-s − 1.08·8-s + 0.333·9-s + 0.878·10-s + 1.11·11-s + 0.186·12-s + 1.70·13-s + 1.52·14-s − 0.616·15-s − 0.571·16-s + 0.315·17-s + 0.274·18-s + 0.0958·19-s − 0.345·20-s − 1.07·21-s + 0.916·22-s − 1.52·23-s + 0.628·24-s + 0.140·25-s + 1.40·26-s − 0.192·27-s − 0.600·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(4.344578749\)
\(L(\frac12)\) \(\approx\) \(4.344578749\)
\(L(\frac{11}{2})\) 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 \( 1 - 18.6T + 512T^{2} \)
5 \( 1 - 1.49e3T + 1.95e6T^{2} \)
7 \( 1 - 1.18e4T + 4.03e7T^{2} \)
11 \( 1 - 5.41e4T + 2.35e9T^{2} \)
13 \( 1 - 1.76e5T + 1.06e10T^{2} \)
17 \( 1 - 1.08e5T + 1.18e11T^{2} \)
19 \( 1 - 5.44e4T + 3.22e11T^{2} \)
23 \( 1 + 2.05e6T + 1.80e12T^{2} \)
29 \( 1 + 3.15e6T + 1.45e13T^{2} \)
31 \( 1 - 5.67e6T + 2.64e13T^{2} \)
37 \( 1 - 1.48e7T + 1.29e14T^{2} \)
41 \( 1 + 2.73e7T + 3.27e14T^{2} \)
43 \( 1 - 1.96e7T + 5.02e14T^{2} \)
47 \( 1 + 1.60e7T + 1.11e15T^{2} \)
53 \( 1 + 4.91e7T + 3.29e15T^{2} \)
61 \( 1 - 9.09e7T + 1.16e16T^{2} \)
67 \( 1 + 1.83e7T + 2.72e16T^{2} \)
71 \( 1 + 3.08e8T + 4.58e16T^{2} \)
73 \( 1 - 1.21e7T + 5.88e16T^{2} \)
79 \( 1 + 2.20e8T + 1.19e17T^{2} \)
83 \( 1 - 4.48e8T + 1.86e17T^{2} \)
89 \( 1 - 7.18e8T + 3.50e17T^{2} \)
97 \( 1 + 1.57e9T + 7.60e17T^{2} \)
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

−11.38950425191940642608879118122, −10.12383527535197030702999300813, −8.985442195050720738117420299798, −8.059797125964045788956523195078, −6.23250783515934606918527121600, −5.75286280967338834327491120143, −4.66274349987443231142880820966, −3.80822395614208462352118407867, −1.86654002683893347293371294864, −1.05499059598594061148343710262, 1.05499059598594061148343710262, 1.86654002683893347293371294864, 3.80822395614208462352118407867, 4.66274349987443231142880820966, 5.75286280967338834327491120143, 6.23250783515934606918527121600, 8.059797125964045788956523195078, 8.985442195050720738117420299798, 10.12383527535197030702999300813, 11.38950425191940642608879118122

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