Properties

Label 2-177-1.1-c11-0-90
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  + 40.1·2-s − 243·3-s − 437.·4-s + 1.08e4·5-s − 9.75e3·6-s − 2.15e3·7-s − 9.97e4·8-s + 5.90e4·9-s + 4.35e5·10-s − 5.58e5·11-s + 1.06e5·12-s + 1.91e6·13-s − 8.65e4·14-s − 2.63e6·15-s − 3.10e6·16-s + 3.75e6·17-s + 2.36e6·18-s − 1.12e7·19-s − 4.75e6·20-s + 5.24e5·21-s − 2.23e7·22-s − 2.13e7·23-s + 2.42e7·24-s + 6.90e7·25-s + 7.68e7·26-s − 1.43e7·27-s + 9.44e5·28-s + ⋯
L(s)  = 1  + 0.886·2-s − 0.577·3-s − 0.213·4-s + 1.55·5-s − 0.511·6-s − 0.0485·7-s − 1.07·8-s + 0.333·9-s + 1.37·10-s − 1.04·11-s + 0.123·12-s + 1.43·13-s − 0.0430·14-s − 0.897·15-s − 0.740·16-s + 0.640·17-s + 0.295·18-s − 1.03·19-s − 0.332·20-s + 0.0280·21-s − 0.926·22-s − 0.692·23-s + 0.621·24-s + 1.41·25-s + 1.26·26-s − 0.192·27-s + 0.0103·28-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 \( 1 - 40.1T + 2.04e3T^{2} \)
5 \( 1 - 1.08e4T + 4.88e7T^{2} \)
7 \( 1 + 2.15e3T + 1.97e9T^{2} \)
11 \( 1 + 5.58e5T + 2.85e11T^{2} \)
13 \( 1 - 1.91e6T + 1.79e12T^{2} \)
17 \( 1 - 3.75e6T + 3.42e13T^{2} \)
19 \( 1 + 1.12e7T + 1.16e14T^{2} \)
23 \( 1 + 2.13e7T + 9.52e14T^{2} \)
29 \( 1 + 6.39e7T + 1.22e16T^{2} \)
31 \( 1 + 1.05e7T + 2.54e16T^{2} \)
37 \( 1 - 1.08e8T + 1.77e17T^{2} \)
41 \( 1 - 4.74e8T + 5.50e17T^{2} \)
43 \( 1 + 2.55e8T + 9.29e17T^{2} \)
47 \( 1 + 1.53e8T + 2.47e18T^{2} \)
53 \( 1 - 1.48e8T + 9.26e18T^{2} \)
61 \( 1 + 5.86e9T + 4.35e19T^{2} \)
67 \( 1 - 2.74e9T + 1.22e20T^{2} \)
71 \( 1 - 1.81e10T + 2.31e20T^{2} \)
73 \( 1 + 1.43e9T + 3.13e20T^{2} \)
79 \( 1 + 1.94e10T + 7.47e20T^{2} \)
83 \( 1 + 2.81e10T + 1.28e21T^{2} \)
89 \( 1 + 1.07e10T + 2.77e21T^{2} \)
97 \( 1 + 2.86e10T + 7.15e21T^{2} \)
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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

−10.24133105356703774476444400570, −9.354997709451856074590449886349, −8.220615928966522624852682370126, −6.42501292098782524920092353840, −5.83621794956066600265449695319, −5.16214224709590773071383205535, −3.92876253141981113412555710678, −2.62451177212707462708407617829, −1.42210196764692269943719539655, 0, 1.42210196764692269943719539655, 2.62451177212707462708407617829, 3.92876253141981113412555710678, 5.16214224709590773071383205535, 5.83621794956066600265449695319, 6.42501292098782524920092353840, 8.220615928966522624852682370126, 9.354997709451856074590449886349, 10.24133105356703774476444400570

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