Properties

Label 2-177-1.1-c9-0-25
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  − 36.7·2-s − 81·3-s + 840.·4-s + 2.30e3·5-s + 2.97e3·6-s − 1.50e3·7-s − 1.20e4·8-s + 6.56e3·9-s − 8.47e4·10-s + 3.54e4·11-s − 6.81e4·12-s − 1.17e5·13-s + 5.55e4·14-s − 1.86e5·15-s + 1.42e4·16-s + 4.66e5·17-s − 2.41e5·18-s − 9.13e4·19-s + 1.93e6·20-s + 1.22e5·21-s − 1.30e6·22-s + 5.07e5·23-s + 9.79e5·24-s + 3.36e6·25-s + 4.32e6·26-s − 5.31e5·27-s − 1.26e6·28-s + ⋯
L(s)  = 1  − 1.62·2-s − 0.577·3-s + 1.64·4-s + 1.64·5-s + 0.938·6-s − 0.237·7-s − 1.04·8-s + 0.333·9-s − 2.68·10-s + 0.730·11-s − 0.948·12-s − 1.14·13-s + 0.386·14-s − 0.952·15-s + 0.0543·16-s + 1.35·17-s − 0.541·18-s − 0.160·19-s + 2.70·20-s + 0.137·21-s − 1.18·22-s + 0.377·23-s + 0.602·24-s + 1.72·25-s + 1.85·26-s − 0.192·27-s − 0.390·28-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

\(L(5)\) \(\approx\) \(1.166494532\)
\(L(\frac12)\) \(\approx\) \(1.166494532\)
\(L(\frac{11}{2})\) 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 \( 1 + 36.7T + 512T^{2} \)
5 \( 1 - 2.30e3T + 1.95e6T^{2} \)
7 \( 1 + 1.50e3T + 4.03e7T^{2} \)
11 \( 1 - 3.54e4T + 2.35e9T^{2} \)
13 \( 1 + 1.17e5T + 1.06e10T^{2} \)
17 \( 1 - 4.66e5T + 1.18e11T^{2} \)
19 \( 1 + 9.13e4T + 3.22e11T^{2} \)
23 \( 1 - 5.07e5T + 1.80e12T^{2} \)
29 \( 1 - 4.06e6T + 1.45e13T^{2} \)
31 \( 1 - 3.34e6T + 2.64e13T^{2} \)
37 \( 1 - 1.50e7T + 1.29e14T^{2} \)
41 \( 1 + 1.13e7T + 3.27e14T^{2} \)
43 \( 1 - 5.74e6T + 5.02e14T^{2} \)
47 \( 1 + 3.17e7T + 1.11e15T^{2} \)
53 \( 1 + 6.11e6T + 3.29e15T^{2} \)
61 \( 1 + 2.97e7T + 1.16e16T^{2} \)
67 \( 1 - 2.83e8T + 2.72e16T^{2} \)
71 \( 1 - 2.32e7T + 4.58e16T^{2} \)
73 \( 1 + 2.01e8T + 5.88e16T^{2} \)
79 \( 1 - 1.98e8T + 1.19e17T^{2} \)
83 \( 1 - 6.95e8T + 1.86e17T^{2} \)
89 \( 1 + 3.82e8T + 3.50e17T^{2} \)
97 \( 1 + 1.05e9T + 7.60e17T^{2} \)
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

−10.50947758691692415850235697748, −9.754954560647851486147624146745, −9.489793833032587523530920403071, −8.139724958291314311149640085063, −6.88985562961144276442043067158, −6.17781322505133278099662621396, −4.96565897976513819419421350565, −2.69978385182041224872904863915, −1.57942228328585620433987897974, −0.74950179864418787533749655292, 0.74950179864418787533749655292, 1.57942228328585620433987897974, 2.69978385182041224872904863915, 4.96565897976513819419421350565, 6.17781322505133278099662621396, 6.88985562961144276442043067158, 8.139724958291314311149640085063, 9.489793833032587523530920403071, 9.754954560647851486147624146745, 10.50947758691692415850235697748

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