Properties

Label 2-177-1.1-c11-0-47
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  − 54.1·2-s + 243·3-s + 884.·4-s − 9.49e3·5-s − 1.31e4·6-s − 2.10e4·7-s + 6.29e4·8-s + 5.90e4·9-s + 5.14e5·10-s − 8.43e5·11-s + 2.14e5·12-s + 2.09e6·13-s + 1.14e6·14-s − 2.30e6·15-s − 5.22e6·16-s − 5.90e6·17-s − 3.19e6·18-s + 4.14e6·19-s − 8.40e6·20-s − 5.12e6·21-s + 4.56e7·22-s − 4.05e7·23-s + 1.53e7·24-s + 4.13e7·25-s − 1.13e8·26-s + 1.43e7·27-s − 1.86e7·28-s + ⋯
L(s)  = 1  − 1.19·2-s + 0.577·3-s + 0.431·4-s − 1.35·5-s − 0.690·6-s − 0.474·7-s + 0.679·8-s + 0.333·9-s + 1.62·10-s − 1.57·11-s + 0.249·12-s + 1.56·13-s + 0.567·14-s − 0.784·15-s − 1.24·16-s − 1.00·17-s − 0.398·18-s + 0.384·19-s − 0.587·20-s − 0.273·21-s + 1.89·22-s − 1.31·23-s + 0.392·24-s + 0.847·25-s − 1.87·26-s + 0.192·27-s − 0.204·28-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 \( 1 + 54.1T + 2.04e3T^{2} \)
5 \( 1 + 9.49e3T + 4.88e7T^{2} \)
7 \( 1 + 2.10e4T + 1.97e9T^{2} \)
11 \( 1 + 8.43e5T + 2.85e11T^{2} \)
13 \( 1 - 2.09e6T + 1.79e12T^{2} \)
17 \( 1 + 5.90e6T + 3.42e13T^{2} \)
19 \( 1 - 4.14e6T + 1.16e14T^{2} \)
23 \( 1 + 4.05e7T + 9.52e14T^{2} \)
29 \( 1 - 7.14e7T + 1.22e16T^{2} \)
31 \( 1 - 2.22e8T + 2.54e16T^{2} \)
37 \( 1 - 5.50e8T + 1.77e17T^{2} \)
41 \( 1 + 3.58e8T + 5.50e17T^{2} \)
43 \( 1 + 8.90e8T + 9.29e17T^{2} \)
47 \( 1 - 2.37e9T + 2.47e18T^{2} \)
53 \( 1 - 2.90e9T + 9.26e18T^{2} \)
61 \( 1 + 7.97e9T + 4.35e19T^{2} \)
67 \( 1 - 9.55e9T + 1.22e20T^{2} \)
71 \( 1 + 1.49e9T + 2.31e20T^{2} \)
73 \( 1 + 4.31e8T + 3.13e20T^{2} \)
79 \( 1 - 4.22e10T + 7.47e20T^{2} \)
83 \( 1 - 6.44e9T + 1.28e21T^{2} \)
89 \( 1 - 1.02e11T + 2.77e21T^{2} \)
97 \( 1 - 1.09e11T + 7.15e21T^{2} \)
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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

−10.12393235259357006448592871963, −8.912041170078662463340774567607, −8.110650881281960858503073423069, −7.76355043030552786649198460456, −6.43814303466903670187498453144, −4.61152139179564486019194197840, −3.63023537485281250017744765250, −2.39541744424349480309844862935, −0.877397295172371023603587502067, 0, 0.877397295172371023603587502067, 2.39541744424349480309844862935, 3.63023537485281250017744765250, 4.61152139179564486019194197840, 6.43814303466903670187498453144, 7.76355043030552786649198460456, 8.110650881281960858503073423069, 8.912041170078662463340774567607, 10.12393235259357006448592871963

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