Properties

Label 2-177-1.1-c11-0-35
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  − 33.4·2-s + 243·3-s − 927.·4-s − 1.18e4·5-s − 8.13e3·6-s + 5.53e4·7-s + 9.96e4·8-s + 5.90e4·9-s + 3.95e5·10-s + 3.10e5·11-s − 2.25e5·12-s + 6.35e5·13-s − 1.85e6·14-s − 2.87e6·15-s − 1.43e6·16-s + 4.94e6·17-s − 1.97e6·18-s + 7.38e6·19-s + 1.09e7·20-s + 1.34e7·21-s − 1.03e7·22-s + 5.49e7·23-s + 2.42e7·24-s + 9.10e7·25-s − 2.12e7·26-s + 1.43e7·27-s − 5.12e7·28-s + ⋯
L(s)  = 1  − 0.739·2-s + 0.577·3-s − 0.452·4-s − 1.69·5-s − 0.427·6-s + 1.24·7-s + 1.07·8-s + 0.333·9-s + 1.25·10-s + 0.580·11-s − 0.261·12-s + 0.474·13-s − 0.919·14-s − 0.977·15-s − 0.341·16-s + 0.845·17-s − 0.246·18-s + 0.684·19-s + 0.766·20-s + 0.718·21-s − 0.429·22-s + 1.78·23-s + 0.620·24-s + 1.86·25-s − 0.350·26-s + 0.192·27-s − 0.563·28-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

\(L(6)\) \(\approx\) \(1.593941299\)
\(L(\frac12)\) \(\approx\) \(1.593941299\)
\(L(\frac{13}{2})\) 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 \( 1 + 33.4T + 2.04e3T^{2} \)
5 \( 1 + 1.18e4T + 4.88e7T^{2} \)
7 \( 1 - 5.53e4T + 1.97e9T^{2} \)
11 \( 1 - 3.10e5T + 2.85e11T^{2} \)
13 \( 1 - 6.35e5T + 1.79e12T^{2} \)
17 \( 1 - 4.94e6T + 3.42e13T^{2} \)
19 \( 1 - 7.38e6T + 1.16e14T^{2} \)
23 \( 1 - 5.49e7T + 9.52e14T^{2} \)
29 \( 1 + 1.08e8T + 1.22e16T^{2} \)
31 \( 1 + 1.74e8T + 2.54e16T^{2} \)
37 \( 1 - 2.79e8T + 1.77e17T^{2} \)
41 \( 1 - 2.62e8T + 5.50e17T^{2} \)
43 \( 1 + 6.62e8T + 9.29e17T^{2} \)
47 \( 1 - 2.01e9T + 2.47e18T^{2} \)
53 \( 1 - 1.32e9T + 9.26e18T^{2} \)
61 \( 1 + 1.28e10T + 4.35e19T^{2} \)
67 \( 1 + 3.98e9T + 1.22e20T^{2} \)
71 \( 1 - 4.51e9T + 2.31e20T^{2} \)
73 \( 1 + 1.35e10T + 3.13e20T^{2} \)
79 \( 1 - 3.46e10T + 7.47e20T^{2} \)
83 \( 1 - 3.01e10T + 1.28e21T^{2} \)
89 \( 1 - 2.22e10T + 2.77e21T^{2} \)
97 \( 1 + 8.94e10T + 7.15e21T^{2} \)
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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

−10.79101821785775758984229264162, −9.263446186149547931939176457434, −8.635467469814692173709797829916, −7.69418488947505187028808835567, −7.37936955011693033815208270266, −5.12180375982232852596211864983, −4.18201094722301421113370756040, −3.35393173374231250147332843626, −1.45513549032921445203798128304, −0.71262934402515369160505182631, 0.71262934402515369160505182631, 1.45513549032921445203798128304, 3.35393173374231250147332843626, 4.18201094722301421113370756040, 5.12180375982232852596211864983, 7.37936955011693033815208270266, 7.69418488947505187028808835567, 8.635467469814692173709797829916, 9.263446186149547931939176457434, 10.79101821785775758984229264162

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