Properties

Label 2-177-1.1-c11-0-45
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  − 68.1·2-s − 243·3-s + 2.59e3·4-s − 1.23e4·5-s + 1.65e4·6-s + 4.70e4·7-s − 3.75e4·8-s + 5.90e4·9-s + 8.42e5·10-s + 9.95e5·11-s − 6.31e5·12-s − 2.04e6·13-s − 3.20e6·14-s + 3.00e6·15-s − 2.76e6·16-s + 4.93e6·17-s − 4.02e6·18-s − 7.46e6·19-s − 3.21e7·20-s − 1.14e7·21-s − 6.78e7·22-s − 6.09e7·23-s + 9.13e6·24-s + 1.03e8·25-s + 1.39e8·26-s − 1.43e7·27-s + 1.22e8·28-s + ⋯
L(s)  = 1  − 1.50·2-s − 0.577·3-s + 1.26·4-s − 1.76·5-s + 0.869·6-s + 1.05·7-s − 0.405·8-s + 0.333·9-s + 2.66·10-s + 1.86·11-s − 0.732·12-s − 1.52·13-s − 1.59·14-s + 1.02·15-s − 0.658·16-s + 0.842·17-s − 0.502·18-s − 0.691·19-s − 2.24·20-s − 0.610·21-s − 2.80·22-s − 1.97·23-s + 0.234·24-s + 2.12·25-s + 2.29·26-s − 0.192·27-s + 1.34·28-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 \( 1 + 68.1T + 2.04e3T^{2} \)
5 \( 1 + 1.23e4T + 4.88e7T^{2} \)
7 \( 1 - 4.70e4T + 1.97e9T^{2} \)
11 \( 1 - 9.95e5T + 2.85e11T^{2} \)
13 \( 1 + 2.04e6T + 1.79e12T^{2} \)
17 \( 1 - 4.93e6T + 3.42e13T^{2} \)
19 \( 1 + 7.46e6T + 1.16e14T^{2} \)
23 \( 1 + 6.09e7T + 9.52e14T^{2} \)
29 \( 1 + 1.21e7T + 1.22e16T^{2} \)
31 \( 1 + 9.81e7T + 2.54e16T^{2} \)
37 \( 1 - 5.10e8T + 1.77e17T^{2} \)
41 \( 1 - 9.11e8T + 5.50e17T^{2} \)
43 \( 1 + 5.23e8T + 9.29e17T^{2} \)
47 \( 1 + 2.51e9T + 2.47e18T^{2} \)
53 \( 1 - 2.99e9T + 9.26e18T^{2} \)
61 \( 1 + 5.25e9T + 4.35e19T^{2} \)
67 \( 1 - 4.16e9T + 1.22e20T^{2} \)
71 \( 1 + 2.59e9T + 2.31e20T^{2} \)
73 \( 1 + 5.20e9T + 3.13e20T^{2} \)
79 \( 1 - 1.63e10T + 7.47e20T^{2} \)
83 \( 1 - 5.97e10T + 1.28e21T^{2} \)
89 \( 1 - 8.53e10T + 2.77e21T^{2} \)
97 \( 1 - 9.77e10T + 7.15e21T^{2} \)
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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

−10.10836091302673282373340987201, −9.097918926169929554316227627796, −7.925550805157353432388336388337, −7.65040909261855035903802177248, −6.52158031886649141715005309644, −4.67269804952743456379715480757, −3.92012143673864871050975369911, −1.91978065685610114640111385908, −0.861470039693743604838652614300, 0, 0.861470039693743604838652614300, 1.91978065685610114640111385908, 3.92012143673864871050975369911, 4.67269804952743456379715480757, 6.52158031886649141715005309644, 7.65040909261855035903802177248, 7.925550805157353432388336388337, 9.097918926169929554316227627796, 10.10836091302673282373340987201

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