Properties

Label 2-177-1.1-c5-0-6
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.68·2-s + 9·3-s − 24.7·4-s − 83.5·5-s + 24.2·6-s − 48.3·7-s − 152.·8-s + 81·9-s − 224.·10-s + 19.9·11-s − 222.·12-s + 611.·13-s − 130.·14-s − 751.·15-s + 381.·16-s + 1.52e3·17-s + 217.·18-s − 1.27e3·19-s + 2.06e3·20-s − 435.·21-s + 53.7·22-s + 3.30e3·23-s − 1.37e3·24-s + 3.84e3·25-s + 1.64e3·26-s + 729·27-s + 1.19e3·28-s + ⋯
L(s)  = 1  + 0.475·2-s + 0.577·3-s − 0.773·4-s − 1.49·5-s + 0.274·6-s − 0.372·7-s − 0.843·8-s + 0.333·9-s − 0.710·10-s + 0.0498·11-s − 0.446·12-s + 1.00·13-s − 0.177·14-s − 0.862·15-s + 0.373·16-s + 1.28·17-s + 0.158·18-s − 0.808·19-s + 1.15·20-s − 0.215·21-s + 0.0236·22-s + 1.30·23-s − 0.486·24-s + 1.23·25-s + 0.477·26-s + 0.192·27-s + 0.288·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.660761036\)
\(L(\frac12)\) \(\approx\) \(1.660761036\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
59 \( 1 + 3.48e3T \)
good2 \( 1 - 2.68T + 32T^{2} \)
5 \( 1 + 83.5T + 3.12e3T^{2} \)
7 \( 1 + 48.3T + 1.68e4T^{2} \)
11 \( 1 - 19.9T + 1.61e5T^{2} \)
13 \( 1 - 611.T + 3.71e5T^{2} \)
17 \( 1 - 1.52e3T + 1.41e6T^{2} \)
19 \( 1 + 1.27e3T + 2.47e6T^{2} \)
23 \( 1 - 3.30e3T + 6.43e6T^{2} \)
29 \( 1 + 2.17e3T + 2.05e7T^{2} \)
31 \( 1 - 7.16e3T + 2.86e7T^{2} \)
37 \( 1 + 5.90e3T + 6.93e7T^{2} \)
41 \( 1 - 2.58e3T + 1.15e8T^{2} \)
43 \( 1 + 5.33e3T + 1.47e8T^{2} \)
47 \( 1 + 1.05e4T + 2.29e8T^{2} \)
53 \( 1 - 6.04e3T + 4.18e8T^{2} \)
61 \( 1 - 134.T + 8.44e8T^{2} \)
67 \( 1 - 4.98e4T + 1.35e9T^{2} \)
71 \( 1 - 6.47e4T + 1.80e9T^{2} \)
73 \( 1 - 4.36e4T + 2.07e9T^{2} \)
79 \( 1 - 7.48e4T + 3.07e9T^{2} \)
83 \( 1 - 5.27e3T + 3.93e9T^{2} \)
89 \( 1 - 8.54e4T + 5.58e9T^{2} \)
97 \( 1 - 4.73e4T + 8.58e9T^{2} \)
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

−12.06227576188018370867229600531, −10.93903361686578287070498589384, −9.632720324576279990524393285943, −8.542212768814982843045226158648, −7.955181282823108552249359886338, −6.56671850162300701319034474384, −4.99809879787258111609107788425, −3.82604256304045866174365312928, −3.25187533875933838619423325117, −0.75301853343308106950638934007, 0.75301853343308106950638934007, 3.25187533875933838619423325117, 3.82604256304045866174365312928, 4.99809879787258111609107788425, 6.56671850162300701319034474384, 7.955181282823108552249359886338, 8.542212768814982843045226158648, 9.632720324576279990524393285943, 10.93903361686578287070498589384, 12.06227576188018370867229600531

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