Properties

Label 2-177-1.1-c11-0-48
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  − 86.7·2-s + 243·3-s + 5.47e3·4-s − 1.17e4·5-s − 2.10e4·6-s + 2.42e4·7-s − 2.97e5·8-s + 5.90e4·9-s + 1.01e6·10-s − 2.25e5·11-s + 1.33e6·12-s − 1.93e6·13-s − 2.10e6·14-s − 2.84e6·15-s + 1.45e7·16-s − 7.58e6·17-s − 5.12e6·18-s + 2.02e7·19-s − 6.42e7·20-s + 5.88e6·21-s + 1.95e7·22-s + 4.13e6·23-s − 7.22e7·24-s + 8.86e7·25-s + 1.67e8·26-s + 1.43e7·27-s + 1.32e8·28-s + ⋯
L(s)  = 1  − 1.91·2-s + 0.577·3-s + 2.67·4-s − 1.67·5-s − 1.10·6-s + 0.545·7-s − 3.20·8-s + 0.333·9-s + 3.21·10-s − 0.422·11-s + 1.54·12-s − 1.44·13-s − 1.04·14-s − 0.968·15-s + 3.47·16-s − 1.29·17-s − 0.638·18-s + 1.87·19-s − 4.48·20-s + 0.314·21-s + 0.809·22-s + 0.133·23-s − 1.85·24-s + 1.81·25-s + 2.77·26-s + 0.192·27-s + 1.45·28-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 \( 1 + 86.7T + 2.04e3T^{2} \)
5 \( 1 + 1.17e4T + 4.88e7T^{2} \)
7 \( 1 - 2.42e4T + 1.97e9T^{2} \)
11 \( 1 + 2.25e5T + 2.85e11T^{2} \)
13 \( 1 + 1.93e6T + 1.79e12T^{2} \)
17 \( 1 + 7.58e6T + 3.42e13T^{2} \)
19 \( 1 - 2.02e7T + 1.16e14T^{2} \)
23 \( 1 - 4.13e6T + 9.52e14T^{2} \)
29 \( 1 - 1.28e8T + 1.22e16T^{2} \)
31 \( 1 + 1.26e8T + 2.54e16T^{2} \)
37 \( 1 - 1.58e8T + 1.77e17T^{2} \)
41 \( 1 - 7.42e8T + 5.50e17T^{2} \)
43 \( 1 + 1.64e8T + 9.29e17T^{2} \)
47 \( 1 + 1.37e9T + 2.47e18T^{2} \)
53 \( 1 - 4.06e8T + 9.26e18T^{2} \)
61 \( 1 - 1.25e9T + 4.35e19T^{2} \)
67 \( 1 + 3.89e9T + 1.22e20T^{2} \)
71 \( 1 - 2.09e10T + 2.31e20T^{2} \)
73 \( 1 - 1.32e10T + 3.13e20T^{2} \)
79 \( 1 - 3.04e10T + 7.47e20T^{2} \)
83 \( 1 + 6.10e10T + 1.28e21T^{2} \)
89 \( 1 - 6.05e10T + 2.77e21T^{2} \)
97 \( 1 + 1.06e11T + 7.15e21T^{2} \)
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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

−9.921883749467526801760611030172, −9.025869057921686846286279202838, −8.064544655975542191067449896984, −7.60024715538521404431294861409, −6.91123207399527314966720033863, −4.80855938148107968153974762673, −3.22126171104857069657859216125, −2.27434734708961379901365630836, −0.896759049382490569846956248121, 0, 0.896759049382490569846956248121, 2.27434734708961379901365630836, 3.22126171104857069657859216125, 4.80855938148107968153974762673, 6.91123207399527314966720033863, 7.60024715538521404431294861409, 8.064544655975542191067449896984, 9.025869057921686846286279202838, 9.921883749467526801760611030172

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