Properties

Label 2-177-1.1-c11-0-12
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 52.1·2-s − 243·3-s + 676.·4-s − 195.·5-s − 1.26e4·6-s − 3.52e4·7-s − 7.15e4·8-s + 5.90e4·9-s − 1.01e4·10-s − 6.66e5·11-s − 1.64e5·12-s − 1.14e6·13-s − 1.83e6·14-s + 4.74e4·15-s − 5.12e6·16-s − 1.54e6·17-s + 3.08e6·18-s − 1.21e7·19-s − 1.32e5·20-s + 8.56e6·21-s − 3.47e7·22-s + 5.18e7·23-s + 1.73e7·24-s − 4.87e7·25-s − 6.00e7·26-s − 1.43e7·27-s − 2.38e7·28-s + ⋯
L(s)  = 1  + 1.15·2-s − 0.577·3-s + 0.330·4-s − 0.0279·5-s − 0.665·6-s − 0.792·7-s − 0.772·8-s + 0.333·9-s − 0.0322·10-s − 1.24·11-s − 0.190·12-s − 0.858·13-s − 0.913·14-s + 0.0161·15-s − 1.22·16-s − 0.264·17-s + 0.384·18-s − 1.12·19-s − 0.00923·20-s + 0.457·21-s − 1.43·22-s + 1.68·23-s + 0.446·24-s − 0.999·25-s − 0.990·26-s − 0.192·27-s − 0.261·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.8703717653\)
\(L(\frac12)\) \(\approx\) \(0.8703717653\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
59 \( 1 + 7.14e8T \)
good2 \( 1 - 52.1T + 2.04e3T^{2} \)
5 \( 1 + 195.T + 4.88e7T^{2} \)
7 \( 1 + 3.52e4T + 1.97e9T^{2} \)
11 \( 1 + 6.66e5T + 2.85e11T^{2} \)
13 \( 1 + 1.14e6T + 1.79e12T^{2} \)
17 \( 1 + 1.54e6T + 3.42e13T^{2} \)
19 \( 1 + 1.21e7T + 1.16e14T^{2} \)
23 \( 1 - 5.18e7T + 9.52e14T^{2} \)
29 \( 1 - 1.11e8T + 1.22e16T^{2} \)
31 \( 1 + 4.07e7T + 2.54e16T^{2} \)
37 \( 1 + 5.91e8T + 1.77e17T^{2} \)
41 \( 1 - 1.38e8T + 5.50e17T^{2} \)
43 \( 1 - 1.48e8T + 9.29e17T^{2} \)
47 \( 1 - 1.81e8T + 2.47e18T^{2} \)
53 \( 1 + 2.53e8T + 9.26e18T^{2} \)
61 \( 1 - 5.28e9T + 4.35e19T^{2} \)
67 \( 1 + 8.08e9T + 1.22e20T^{2} \)
71 \( 1 - 1.55e10T + 2.31e20T^{2} \)
73 \( 1 + 3.19e9T + 3.13e20T^{2} \)
79 \( 1 + 4.22e10T + 7.47e20T^{2} \)
83 \( 1 + 1.40e9T + 1.28e21T^{2} \)
89 \( 1 - 2.78e10T + 2.77e21T^{2} \)
97 \( 1 - 6.20e10T + 7.15e21T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.80871103816256967259908422208, −9.880984040234824319599349543756, −8.699061170209120232321885660047, −7.20882029859473004374691674811, −6.27614293634765485423975544626, −5.27165764301910058577776685327, −4.56309257413059896903633759346, −3.29546260244376146089066522057, −2.33504400390190573750213419379, −0.34014310962195574558658557446, 0.34014310962195574558658557446, 2.33504400390190573750213419379, 3.29546260244376146089066522057, 4.56309257413059896903633759346, 5.27165764301910058577776685327, 6.27614293634765485423975544626, 7.20882029859473004374691674811, 8.699061170209120232321885660047, 9.880984040234824319599349543756, 10.80871103816256967259908422208

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