Properties

Label 2-177-1.1-c11-0-53
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  − 66.8·2-s − 243·3-s + 2.42e3·4-s − 5.68e3·5-s + 1.62e4·6-s + 2.66e4·7-s − 2.53e4·8-s + 5.90e4·9-s + 3.80e5·10-s + 2.82e5·11-s − 5.89e5·12-s + 1.28e6·13-s − 1.78e6·14-s + 1.38e6·15-s − 3.27e6·16-s − 4.91e6·17-s − 3.95e6·18-s − 4.23e6·19-s − 1.38e7·20-s − 6.48e6·21-s − 1.89e7·22-s + 3.04e7·23-s + 6.17e6·24-s − 1.64e7·25-s − 8.60e7·26-s − 1.43e7·27-s + 6.47e7·28-s + ⋯
L(s)  = 1  − 1.47·2-s − 0.577·3-s + 1.18·4-s − 0.813·5-s + 0.853·6-s + 0.600·7-s − 0.274·8-s + 0.333·9-s + 1.20·10-s + 0.529·11-s − 0.684·12-s + 0.960·13-s − 0.887·14-s + 0.469·15-s − 0.780·16-s − 0.838·17-s − 0.492·18-s − 0.392·19-s − 0.964·20-s − 0.346·21-s − 0.783·22-s + 0.987·23-s + 0.158·24-s − 0.337·25-s − 1.41·26-s − 0.192·27-s + 0.711·28-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 \( 1 + 66.8T + 2.04e3T^{2} \)
5 \( 1 + 5.68e3T + 4.88e7T^{2} \)
7 \( 1 - 2.66e4T + 1.97e9T^{2} \)
11 \( 1 - 2.82e5T + 2.85e11T^{2} \)
13 \( 1 - 1.28e6T + 1.79e12T^{2} \)
17 \( 1 + 4.91e6T + 3.42e13T^{2} \)
19 \( 1 + 4.23e6T + 1.16e14T^{2} \)
23 \( 1 - 3.04e7T + 9.52e14T^{2} \)
29 \( 1 - 1.00e8T + 1.22e16T^{2} \)
31 \( 1 + 1.83e8T + 2.54e16T^{2} \)
37 \( 1 + 4.83e8T + 1.77e17T^{2} \)
41 \( 1 + 4.31e8T + 5.50e17T^{2} \)
43 \( 1 + 3.19e8T + 9.29e17T^{2} \)
47 \( 1 - 1.15e9T + 2.47e18T^{2} \)
53 \( 1 + 1.32e9T + 9.26e18T^{2} \)
61 \( 1 - 4.52e9T + 4.35e19T^{2} \)
67 \( 1 - 1.21e10T + 1.22e20T^{2} \)
71 \( 1 - 2.63e10T + 2.31e20T^{2} \)
73 \( 1 + 1.30e9T + 3.13e20T^{2} \)
79 \( 1 + 1.29e10T + 7.47e20T^{2} \)
83 \( 1 + 6.28e10T + 1.28e21T^{2} \)
89 \( 1 - 4.23e10T + 2.77e21T^{2} \)
97 \( 1 - 5.43e10T + 7.15e21T^{2} \)
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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

−10.21246340239437061390579818226, −8.934634902245654671116127132656, −8.380222996094063400039242821503, −7.30097742301345864351994089713, −6.47391004468796812286356737561, −4.89000094807584233338552128890, −3.75655844754108068066567664763, −1.90945722649399092868675582823, −0.961430917348354858688027398924, 0, 0.961430917348354858688027398924, 1.90945722649399092868675582823, 3.75655844754108068066567664763, 4.89000094807584233338552128890, 6.47391004468796812286356737561, 7.30097742301345864351994089713, 8.380222996094063400039242821503, 8.934634902245654671116127132656, 10.21246340239437061390579818226

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