Properties

Label 2-177-1.1-c11-0-46
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.6·2-s − 243·3-s − 1.93e3·4-s − 9.08e3·5-s + 2.58e3·6-s + 5.76e4·7-s + 4.23e4·8-s + 5.90e4·9-s + 9.65e4·10-s + 8.14e5·11-s + 4.70e5·12-s + 2.09e6·13-s − 6.12e5·14-s + 2.20e6·15-s + 3.51e6·16-s − 4.28e5·17-s − 6.27e5·18-s + 2.07e7·19-s + 1.75e7·20-s − 1.40e7·21-s − 8.65e6·22-s − 2.39e7·23-s − 1.02e7·24-s + 3.36e7·25-s − 2.23e7·26-s − 1.43e7·27-s − 1.11e8·28-s + ⋯
L(s)  = 1  − 0.234·2-s − 0.577·3-s − 0.944·4-s − 1.29·5-s + 0.135·6-s + 1.29·7-s + 0.456·8-s + 0.333·9-s + 0.305·10-s + 1.52·11-s + 0.545·12-s + 1.56·13-s − 0.304·14-s + 0.750·15-s + 0.837·16-s − 0.0732·17-s − 0.0782·18-s + 1.92·19-s + 1.22·20-s − 0.748·21-s − 0.358·22-s − 0.777·23-s − 0.263·24-s + 0.689·25-s − 0.368·26-s − 0.192·27-s − 1.22·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(1.783714336\)
\(L(\frac12)\) \(\approx\) \(1.783714336\)
\(L(\frac{13}{2})\) 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 \( 1 + 10.6T + 2.04e3T^{2} \)
5 \( 1 + 9.08e3T + 4.88e7T^{2} \)
7 \( 1 - 5.76e4T + 1.97e9T^{2} \)
11 \( 1 - 8.14e5T + 2.85e11T^{2} \)
13 \( 1 - 2.09e6T + 1.79e12T^{2} \)
17 \( 1 + 4.28e5T + 3.42e13T^{2} \)
19 \( 1 - 2.07e7T + 1.16e14T^{2} \)
23 \( 1 + 2.39e7T + 9.52e14T^{2} \)
29 \( 1 - 2.13e8T + 1.22e16T^{2} \)
31 \( 1 - 1.63e8T + 2.54e16T^{2} \)
37 \( 1 - 2.02e8T + 1.77e17T^{2} \)
41 \( 1 + 7.88e8T + 5.50e17T^{2} \)
43 \( 1 - 5.39e8T + 9.29e17T^{2} \)
47 \( 1 - 4.74e8T + 2.47e18T^{2} \)
53 \( 1 - 1.49e9T + 9.26e18T^{2} \)
61 \( 1 - 2.95e9T + 4.35e19T^{2} \)
67 \( 1 - 1.73e10T + 1.22e20T^{2} \)
71 \( 1 - 2.37e10T + 2.31e20T^{2} \)
73 \( 1 - 1.89e10T + 3.13e20T^{2} \)
79 \( 1 - 2.68e10T + 7.47e20T^{2} \)
83 \( 1 - 1.55e10T + 1.28e21T^{2} \)
89 \( 1 + 5.66e10T + 2.77e21T^{2} \)
97 \( 1 + 7.99e10T + 7.15e21T^{2} \)
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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

−10.88218870283392194423078747427, −9.587778865578214312725895451127, −8.393433361890738216093368170589, −7.996941842568837174281739198217, −6.62283925346055342373959375212, −5.23142313473153012023287445359, −4.25391372402759609890217067569, −3.65280063011739007376845142182, −1.16971596317183510825801741323, −0.876830350430755677496598983982, 0.876830350430755677496598983982, 1.16971596317183510825801741323, 3.65280063011739007376845142182, 4.25391372402759609890217067569, 5.23142313473153012023287445359, 6.62283925346055342373959375212, 7.996941842568837174281739198217, 8.393433361890738216093368170589, 9.587778865578214312725895451127, 10.88218870283392194423078747427

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