Properties

Label 2-177-1.1-c11-0-40
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  − 45.9·2-s + 243·3-s + 64.3·4-s − 1.19e4·5-s − 1.11e4·6-s − 7.98e4·7-s + 9.11e4·8-s + 5.90e4·9-s + 5.48e5·10-s + 9.46e5·11-s + 1.56e4·12-s − 1.76e6·13-s + 3.67e6·14-s − 2.90e6·15-s − 4.32e6·16-s − 7.18e6·17-s − 2.71e6·18-s − 1.18e7·19-s − 7.68e5·20-s − 1.94e7·21-s − 4.35e7·22-s + 1.07e7·23-s + 2.21e7·24-s + 9.36e7·25-s + 8.11e7·26-s + 1.43e7·27-s − 5.14e6·28-s + ⋯
L(s)  = 1  − 1.01·2-s + 0.577·3-s + 0.0314·4-s − 1.70·5-s − 0.586·6-s − 1.79·7-s + 0.983·8-s + 0.333·9-s + 1.73·10-s + 1.77·11-s + 0.0181·12-s − 1.31·13-s + 1.82·14-s − 0.986·15-s − 1.03·16-s − 1.22·17-s − 0.338·18-s − 1.09·19-s − 0.0537·20-s − 1.03·21-s − 1.80·22-s + 0.346·23-s + 0.567·24-s + 1.91·25-s + 1.33·26-s + 0.192·27-s − 0.0564·28-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 \( 1 + 45.9T + 2.04e3T^{2} \)
5 \( 1 + 1.19e4T + 4.88e7T^{2} \)
7 \( 1 + 7.98e4T + 1.97e9T^{2} \)
11 \( 1 - 9.46e5T + 2.85e11T^{2} \)
13 \( 1 + 1.76e6T + 1.79e12T^{2} \)
17 \( 1 + 7.18e6T + 3.42e13T^{2} \)
19 \( 1 + 1.18e7T + 1.16e14T^{2} \)
23 \( 1 - 1.07e7T + 9.52e14T^{2} \)
29 \( 1 - 1.10e6T + 1.22e16T^{2} \)
31 \( 1 - 1.34e8T + 2.54e16T^{2} \)
37 \( 1 + 4.45e8T + 1.77e17T^{2} \)
41 \( 1 - 1.34e9T + 5.50e17T^{2} \)
43 \( 1 + 5.78e8T + 9.29e17T^{2} \)
47 \( 1 - 2.46e9T + 2.47e18T^{2} \)
53 \( 1 - 3.35e9T + 9.26e18T^{2} \)
61 \( 1 - 2.55e9T + 4.35e19T^{2} \)
67 \( 1 - 1.07e10T + 1.22e20T^{2} \)
71 \( 1 - 1.60e10T + 2.31e20T^{2} \)
73 \( 1 - 2.47e10T + 3.13e20T^{2} \)
79 \( 1 + 3.19e10T + 7.47e20T^{2} \)
83 \( 1 + 3.85e10T + 1.28e21T^{2} \)
89 \( 1 + 7.10e10T + 2.77e21T^{2} \)
97 \( 1 + 6.48e10T + 7.15e21T^{2} \)
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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

−9.801185780545664409787875571333, −9.075251891993690147784593288776, −8.439697127414805781496395027333, −7.11712872995234944847183398438, −6.77570968632734979679535812444, −4.30038675697400554537726995776, −3.86929436967937135740545273903, −2.51770772231349698822446387616, −0.75457120137739441859012090612, 0, 0.75457120137739441859012090612, 2.51770772231349698822446387616, 3.86929436967937135740545273903, 4.30038675697400554537726995776, 6.77570968632734979679535812444, 7.11712872995234944847183398438, 8.439697127414805781496395027333, 9.075251891993690147784593288776, 9.801185780545664409787875571333

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