Properties

Label 2-177-1.1-c9-0-13
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  + 26.7·2-s − 81·3-s + 203.·4-s − 302.·5-s − 2.16e3·6-s − 5.73e3·7-s − 8.24e3·8-s + 6.56e3·9-s − 8.08e3·10-s + 8.93e3·11-s − 1.65e4·12-s − 3.41e4·13-s − 1.53e5·14-s + 2.44e4·15-s − 3.24e5·16-s + 2.51e4·17-s + 1.75e5·18-s + 4.11e4·19-s − 6.15e4·20-s + 4.64e5·21-s + 2.38e5·22-s − 1.88e6·23-s + 6.68e5·24-s − 1.86e6·25-s − 9.13e5·26-s − 5.31e5·27-s − 1.16e6·28-s + ⋯
L(s)  = 1  + 1.18·2-s − 0.577·3-s + 0.397·4-s − 0.216·5-s − 0.682·6-s − 0.903·7-s − 0.711·8-s + 0.333·9-s − 0.255·10-s + 0.183·11-s − 0.229·12-s − 0.331·13-s − 1.06·14-s + 0.124·15-s − 1.23·16-s + 0.0730·17-s + 0.394·18-s + 0.0725·19-s − 0.0860·20-s + 0.521·21-s + 0.217·22-s − 1.40·23-s + 0.411·24-s − 0.953·25-s − 0.392·26-s − 0.192·27-s − 0.359·28-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

\(L(5)\) \(\approx\) \(1.714085912\)
\(L(\frac12)\) \(\approx\) \(1.714085912\)
\(L(\frac{11}{2})\) 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 \( 1 - 26.7T + 512T^{2} \)
5 \( 1 + 302.T + 1.95e6T^{2} \)
7 \( 1 + 5.73e3T + 4.03e7T^{2} \)
11 \( 1 - 8.93e3T + 2.35e9T^{2} \)
13 \( 1 + 3.41e4T + 1.06e10T^{2} \)
17 \( 1 - 2.51e4T + 1.18e11T^{2} \)
19 \( 1 - 4.11e4T + 3.22e11T^{2} \)
23 \( 1 + 1.88e6T + 1.80e12T^{2} \)
29 \( 1 - 4.48e6T + 1.45e13T^{2} \)
31 \( 1 + 2.37e6T + 2.64e13T^{2} \)
37 \( 1 - 1.10e7T + 1.29e14T^{2} \)
41 \( 1 - 2.38e7T + 3.27e14T^{2} \)
43 \( 1 - 3.14e7T + 5.02e14T^{2} \)
47 \( 1 - 2.63e7T + 1.11e15T^{2} \)
53 \( 1 + 2.61e7T + 3.29e15T^{2} \)
61 \( 1 + 1.33e7T + 1.16e16T^{2} \)
67 \( 1 - 1.72e8T + 2.72e16T^{2} \)
71 \( 1 - 1.07e8T + 4.58e16T^{2} \)
73 \( 1 + 4.60e8T + 5.88e16T^{2} \)
79 \( 1 - 2.14e8T + 1.19e17T^{2} \)
83 \( 1 - 3.37e8T + 1.86e17T^{2} \)
89 \( 1 - 1.74e8T + 3.50e17T^{2} \)
97 \( 1 - 8.52e8T + 7.60e17T^{2} \)
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

−11.39980943748854060625903813376, −10.09617017171219885096396390410, −9.201529224680573656524110258617, −7.66663970930937014213263058965, −6.36080057201743636675315381612, −5.78864377996456523627927101837, −4.50855610302522454414691775219, −3.70352854429973052367800605046, −2.45380317004302090345267562769, −0.53306618664080250509451749933, 0.53306618664080250509451749933, 2.45380317004302090345267562769, 3.70352854429973052367800605046, 4.50855610302522454414691775219, 5.78864377996456523627927101837, 6.36080057201743636675315381612, 7.66663970930937014213263058965, 9.201529224680573656524110258617, 10.09617017171219885096396390410, 11.39980943748854060625903813376

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