Properties

Label 2-177-1.1-c11-0-2
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  − 38.5·2-s − 243·3-s − 565.·4-s − 1.32e4·5-s + 9.35e3·6-s − 8.92e3·7-s + 1.00e5·8-s + 5.90e4·9-s + 5.09e5·10-s − 5.48e5·11-s + 1.37e5·12-s + 1.36e6·13-s + 3.43e5·14-s + 3.21e6·15-s − 2.71e6·16-s + 5.04e6·17-s − 2.27e6·18-s − 5.16e6·19-s + 7.47e6·20-s + 2.16e6·21-s + 2.11e7·22-s − 9.48e6·23-s − 2.44e7·24-s + 1.26e8·25-s − 5.26e7·26-s − 1.43e7·27-s + 5.04e6·28-s + ⋯
L(s)  = 1  − 0.850·2-s − 0.577·3-s − 0.276·4-s − 1.89·5-s + 0.491·6-s − 0.200·7-s + 1.08·8-s + 0.333·9-s + 1.61·10-s − 1.02·11-s + 0.159·12-s + 1.02·13-s + 0.170·14-s + 1.09·15-s − 0.647·16-s + 0.862·17-s − 0.283·18-s − 0.478·19-s + 0.522·20-s + 0.115·21-s + 0.874·22-s − 0.307·23-s − 0.626·24-s + 2.58·25-s − 0.869·26-s − 0.192·27-s + 0.0553·28-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

\(L(6)\) \(\approx\) \(0.05624710107\)
\(L(\frac12)\) \(\approx\) \(0.05624710107\)
\(L(\frac{13}{2})\) 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 \( 1 + 38.5T + 2.04e3T^{2} \)
5 \( 1 + 1.32e4T + 4.88e7T^{2} \)
7 \( 1 + 8.92e3T + 1.97e9T^{2} \)
11 \( 1 + 5.48e5T + 2.85e11T^{2} \)
13 \( 1 - 1.36e6T + 1.79e12T^{2} \)
17 \( 1 - 5.04e6T + 3.42e13T^{2} \)
19 \( 1 + 5.16e6T + 1.16e14T^{2} \)
23 \( 1 + 9.48e6T + 9.52e14T^{2} \)
29 \( 1 + 1.29e8T + 1.22e16T^{2} \)
31 \( 1 + 9.27e7T + 2.54e16T^{2} \)
37 \( 1 + 3.97e7T + 1.77e17T^{2} \)
41 \( 1 - 9.65e7T + 5.50e17T^{2} \)
43 \( 1 + 6.50e8T + 9.29e17T^{2} \)
47 \( 1 + 1.60e9T + 2.47e18T^{2} \)
53 \( 1 + 3.48e9T + 9.26e18T^{2} \)
61 \( 1 - 5.15e9T + 4.35e19T^{2} \)
67 \( 1 + 5.22e9T + 1.22e20T^{2} \)
71 \( 1 - 1.67e10T + 2.31e20T^{2} \)
73 \( 1 - 2.13e10T + 3.13e20T^{2} \)
79 \( 1 + 3.48e10T + 7.47e20T^{2} \)
83 \( 1 + 6.02e10T + 1.28e21T^{2} \)
89 \( 1 + 5.21e10T + 2.77e21T^{2} \)
97 \( 1 + 1.33e11T + 7.15e21T^{2} \)
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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

−10.82952228552870270243344536831, −9.686016318254650730113872221744, −8.338403413045853664065022031758, −7.945180340189634649748606108126, −6.95236142034430212258870314756, −5.35266876694770398348563010118, −4.24551134414948220977416453894, −3.38229298947710278299324913100, −1.33931166169332928953185737754, −0.14013715889950874542297128125, 0.14013715889950874542297128125, 1.33931166169332928953185737754, 3.38229298947710278299324913100, 4.24551134414948220977416453894, 5.35266876694770398348563010118, 6.95236142034430212258870314756, 7.945180340189634649748606108126, 8.338403413045853664065022031758, 9.686016318254650730113872221744, 10.82952228552870270243344536831

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