Properties

Label 2-177-1.1-c11-0-73
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 49.4·2-s − 243·3-s + 400.·4-s + 6.60e3·5-s + 1.20e4·6-s − 1.79e4·7-s + 8.15e4·8-s + 5.90e4·9-s − 3.27e5·10-s + 9.78e5·11-s − 9.72e4·12-s + 4.94e5·13-s + 8.87e5·14-s − 1.60e6·15-s − 4.85e6·16-s − 9.53e6·17-s − 2.92e6·18-s + 6.89e6·19-s + 2.64e6·20-s + 4.35e6·21-s − 4.83e7·22-s + 2.50e7·23-s − 1.98e7·24-s − 5.15e6·25-s − 2.44e7·26-s − 1.43e7·27-s − 7.17e6·28-s + ⋯
L(s)  = 1  − 1.09·2-s − 0.577·3-s + 0.195·4-s + 0.945·5-s + 0.631·6-s − 0.403·7-s + 0.879·8-s + 0.333·9-s − 1.03·10-s + 1.83·11-s − 0.112·12-s + 0.369·13-s + 0.440·14-s − 0.546·15-s − 1.15·16-s − 1.62·17-s − 0.364·18-s + 0.639·19-s + 0.184·20-s + 0.232·21-s − 2.00·22-s + 0.811·23-s − 0.507·24-s − 0.105·25-s − 0.403·26-s − 0.192·27-s − 0.0788·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 + 49.4T + 2.04e3T^{2} \)
5 \( 1 - 6.60e3T + 4.88e7T^{2} \)
7 \( 1 + 1.79e4T + 1.97e9T^{2} \)
11 \( 1 - 9.78e5T + 2.85e11T^{2} \)
13 \( 1 - 4.94e5T + 1.79e12T^{2} \)
17 \( 1 + 9.53e6T + 3.42e13T^{2} \)
19 \( 1 - 6.89e6T + 1.16e14T^{2} \)
23 \( 1 - 2.50e7T + 9.52e14T^{2} \)
29 \( 1 + 2.14e8T + 1.22e16T^{2} \)
31 \( 1 - 2.69e8T + 2.54e16T^{2} \)
37 \( 1 + 2.91e8T + 1.77e17T^{2} \)
41 \( 1 + 6.14e8T + 5.50e17T^{2} \)
43 \( 1 + 9.63e8T + 9.29e17T^{2} \)
47 \( 1 + 2.17e9T + 2.47e18T^{2} \)
53 \( 1 - 2.60e9T + 9.26e18T^{2} \)
61 \( 1 - 3.67e9T + 4.35e19T^{2} \)
67 \( 1 - 1.09e10T + 1.22e20T^{2} \)
71 \( 1 + 1.23e10T + 2.31e20T^{2} \)
73 \( 1 - 1.39e10T + 3.13e20T^{2} \)
79 \( 1 - 7.86e9T + 7.47e20T^{2} \)
83 \( 1 - 3.44e10T + 1.28e21T^{2} \)
89 \( 1 - 6.24e10T + 2.77e21T^{2} \)
97 \( 1 + 1.46e11T + 7.15e21T^{2} \)
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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

−9.813941025459650699768103148447, −9.368840823003400404123014145520, −8.508064389570572939267782905405, −6.90339426098773415810687306417, −6.39898788147644892948709688621, −5.01221140577684173179351737036, −3.78557813588529646613022692489, −1.90920349233490090325939991589, −1.16715280343874764232628812553, 0, 1.16715280343874764232628812553, 1.90920349233490090325939991589, 3.78557813588529646613022692489, 5.01221140577684173179351737036, 6.39898788147644892948709688621, 6.90339426098773415810687306417, 8.508064389570572939267782905405, 9.368840823003400404123014145520, 9.813941025459650699768103148447

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