Properties

Label 2-177-1.1-c11-0-71
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  + 38.1·2-s − 243·3-s − 590.·4-s + 118.·5-s − 9.27e3·6-s − 2.49e4·7-s − 1.00e5·8-s + 5.90e4·9-s + 4.51e3·10-s + 7.83e4·11-s + 1.43e5·12-s + 5.16e5·13-s − 9.54e5·14-s − 2.87e4·15-s − 2.63e6·16-s + 7.07e5·17-s + 2.25e6·18-s + 1.93e7·19-s − 6.98e4·20-s + 6.07e6·21-s + 2.98e6·22-s − 4.56e6·23-s + 2.44e7·24-s − 4.88e7·25-s + 1.97e7·26-s − 1.43e7·27-s + 1.47e7·28-s + ⋯
L(s)  = 1  + 0.843·2-s − 0.577·3-s − 0.288·4-s + 0.0169·5-s − 0.487·6-s − 0.562·7-s − 1.08·8-s + 0.333·9-s + 0.0142·10-s + 0.146·11-s + 0.166·12-s + 0.385·13-s − 0.474·14-s − 0.00978·15-s − 0.628·16-s + 0.120·17-s + 0.281·18-s + 1.79·19-s − 0.00488·20-s + 0.324·21-s + 0.123·22-s − 0.147·23-s + 0.627·24-s − 0.999·25-s + 0.325·26-s − 0.192·27-s + 0.162·28-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 \( 1 - 38.1T + 2.04e3T^{2} \)
5 \( 1 - 118.T + 4.88e7T^{2} \)
7 \( 1 + 2.49e4T + 1.97e9T^{2} \)
11 \( 1 - 7.83e4T + 2.85e11T^{2} \)
13 \( 1 - 5.16e5T + 1.79e12T^{2} \)
17 \( 1 - 7.07e5T + 3.42e13T^{2} \)
19 \( 1 - 1.93e7T + 1.16e14T^{2} \)
23 \( 1 + 4.56e6T + 9.52e14T^{2} \)
29 \( 1 - 1.21e8T + 1.22e16T^{2} \)
31 \( 1 - 5.43e7T + 2.54e16T^{2} \)
37 \( 1 - 3.64e8T + 1.77e17T^{2} \)
41 \( 1 + 9.48e7T + 5.50e17T^{2} \)
43 \( 1 + 1.12e9T + 9.29e17T^{2} \)
47 \( 1 + 1.05e9T + 2.47e18T^{2} \)
53 \( 1 - 1.10e9T + 9.26e18T^{2} \)
61 \( 1 - 9.70e9T + 4.35e19T^{2} \)
67 \( 1 + 1.88e10T + 1.22e20T^{2} \)
71 \( 1 + 1.71e9T + 2.31e20T^{2} \)
73 \( 1 + 6.45e9T + 3.13e20T^{2} \)
79 \( 1 - 3.94e9T + 7.47e20T^{2} \)
83 \( 1 - 1.19e10T + 1.28e21T^{2} \)
89 \( 1 - 2.13e10T + 2.77e21T^{2} \)
97 \( 1 + 4.49e10T + 7.15e21T^{2} \)
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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

−10.08486717665929320530251052265, −9.394326353231305728436390247129, −8.083601275578034508340096954703, −6.69958477382988573700915643259, −5.82502906543812463566420440033, −4.96002870120936324723807732712, −3.85032285987182164901737666583, −2.92742522070789488015695941352, −1.14596099144472147764748238629, 0, 1.14596099144472147764748238629, 2.92742522070789488015695941352, 3.85032285987182164901737666583, 4.96002870120936324723807732712, 5.82502906543812463566420440033, 6.69958477382988573700915643259, 8.083601275578034508340096954703, 9.394326353231305728436390247129, 10.08486717665929320530251052265

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