Properties

Label 2-177-1.1-c11-0-16
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  + 24.9·2-s − 243·3-s − 1.42e3·4-s + 1.13e4·5-s − 6.07e3·6-s − 4.61e4·7-s − 8.67e4·8-s + 5.90e4·9-s + 2.83e5·10-s − 1.58e5·11-s + 3.45e5·12-s − 2.30e6·13-s − 1.15e6·14-s − 2.75e6·15-s + 7.47e5·16-s − 3.64e6·17-s + 1.47e6·18-s + 8.79e6·19-s − 1.61e7·20-s + 1.12e7·21-s − 3.97e6·22-s − 4.28e6·23-s + 2.10e7·24-s + 7.99e7·25-s − 5.76e7·26-s − 1.43e7·27-s + 6.56e7·28-s + ⋯
L(s)  = 1  + 0.552·2-s − 0.577·3-s − 0.695·4-s + 1.62·5-s − 0.318·6-s − 1.03·7-s − 0.935·8-s + 0.333·9-s + 0.896·10-s − 0.297·11-s + 0.401·12-s − 1.72·13-s − 0.572·14-s − 0.937·15-s + 0.178·16-s − 0.622·17-s + 0.184·18-s + 0.814·19-s − 1.12·20-s + 0.598·21-s − 0.164·22-s − 0.138·23-s + 0.540·24-s + 1.63·25-s − 0.951·26-s − 0.192·27-s + 0.720·28-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

\(L(6)\) \(\approx\) \(1.258760812\)
\(L(\frac12)\) \(\approx\) \(1.258760812\)
\(L(\frac{13}{2})\) 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 \( 1 - 24.9T + 2.04e3T^{2} \)
5 \( 1 - 1.13e4T + 4.88e7T^{2} \)
7 \( 1 + 4.61e4T + 1.97e9T^{2} \)
11 \( 1 + 1.58e5T + 2.85e11T^{2} \)
13 \( 1 + 2.30e6T + 1.79e12T^{2} \)
17 \( 1 + 3.64e6T + 3.42e13T^{2} \)
19 \( 1 - 8.79e6T + 1.16e14T^{2} \)
23 \( 1 + 4.28e6T + 9.52e14T^{2} \)
29 \( 1 + 9.38e7T + 1.22e16T^{2} \)
31 \( 1 + 1.47e8T + 2.54e16T^{2} \)
37 \( 1 - 3.64e8T + 1.77e17T^{2} \)
41 \( 1 + 1.41e8T + 5.50e17T^{2} \)
43 \( 1 - 2.12e7T + 9.29e17T^{2} \)
47 \( 1 + 9.13e7T + 2.47e18T^{2} \)
53 \( 1 - 2.00e9T + 9.26e18T^{2} \)
61 \( 1 + 3.63e9T + 4.35e19T^{2} \)
67 \( 1 - 7.77e8T + 1.22e20T^{2} \)
71 \( 1 + 2.23e10T + 2.31e20T^{2} \)
73 \( 1 + 2.80e10T + 3.13e20T^{2} \)
79 \( 1 - 4.41e10T + 7.47e20T^{2} \)
83 \( 1 - 2.91e10T + 1.28e21T^{2} \)
89 \( 1 - 2.60e10T + 2.77e21T^{2} \)
97 \( 1 - 4.00e9T + 7.15e21T^{2} \)
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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

−10.37528944435996449280845758305, −9.627095585034066345919934825710, −9.200027939215127865932137443856, −7.29170433613647564239088029553, −6.15538966514523447358643044954, −5.48434374987456647273803323069, −4.65424701906903844925668936543, −3.12823559383491889164432144229, −2.07011620390077984642649580058, −0.46085836384142906950712020296, 0.46085836384142906950712020296, 2.07011620390077984642649580058, 3.12823559383491889164432144229, 4.65424701906903844925668936543, 5.48434374987456647273803323069, 6.15538966514523447358643044954, 7.29170433613647564239088029553, 9.200027939215127865932137443856, 9.627095585034066345919934825710, 10.37528944435996449280845758305

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