Properties

Label 2-177-1.1-c13-0-91
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 162.·2-s + 729·3-s + 1.83e4·4-s + 2.77e4·5-s + 1.18e5·6-s − 5.95e4·7-s + 1.65e6·8-s + 5.31e5·9-s + 4.52e6·10-s − 2.68e6·11-s + 1.33e7·12-s − 3.02e6·13-s − 9.70e6·14-s + 2.02e7·15-s + 1.18e8·16-s + 1.27e8·17-s + 8.65e7·18-s + 3.71e8·19-s + 5.08e8·20-s − 4.34e7·21-s − 4.36e8·22-s − 8.99e8·23-s + 1.20e9·24-s − 4.49e8·25-s − 4.93e8·26-s + 3.87e8·27-s − 1.09e9·28-s + ⋯
L(s)  = 1  + 1.79·2-s + 0.577·3-s + 2.23·4-s + 0.794·5-s + 1.03·6-s − 0.191·7-s + 2.22·8-s + 0.333·9-s + 1.42·10-s − 0.456·11-s + 1.29·12-s − 0.174·13-s − 0.344·14-s + 0.458·15-s + 1.76·16-s + 1.27·17-s + 0.599·18-s + 1.80·19-s + 1.77·20-s − 0.110·21-s − 0.821·22-s − 1.26·23-s + 1.28·24-s − 0.368·25-s − 0.313·26-s + 0.192·27-s − 0.428·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(12.27874079\)
\(L(\frac12)\) \(\approx\) \(12.27874079\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 729T \)
59 \( 1 + 4.21e10T \)
good2 \( 1 - 162.T + 8.19e3T^{2} \)
5 \( 1 - 2.77e4T + 1.22e9T^{2} \)
7 \( 1 + 5.95e4T + 9.68e10T^{2} \)
11 \( 1 + 2.68e6T + 3.45e13T^{2} \)
13 \( 1 + 3.02e6T + 3.02e14T^{2} \)
17 \( 1 - 1.27e8T + 9.90e15T^{2} \)
19 \( 1 - 3.71e8T + 4.20e16T^{2} \)
23 \( 1 + 8.99e8T + 5.04e17T^{2} \)
29 \( 1 - 3.38e9T + 1.02e19T^{2} \)
31 \( 1 - 4.54e9T + 2.44e19T^{2} \)
37 \( 1 + 1.05e10T + 2.43e20T^{2} \)
41 \( 1 + 1.90e10T + 9.25e20T^{2} \)
43 \( 1 - 6.35e10T + 1.71e21T^{2} \)
47 \( 1 - 9.70e10T + 5.46e21T^{2} \)
53 \( 1 - 2.65e11T + 2.60e22T^{2} \)
61 \( 1 - 1.69e11T + 1.61e23T^{2} \)
67 \( 1 - 4.94e11T + 5.48e23T^{2} \)
71 \( 1 - 2.21e10T + 1.16e24T^{2} \)
73 \( 1 - 4.50e11T + 1.67e24T^{2} \)
79 \( 1 + 1.95e12T + 4.66e24T^{2} \)
83 \( 1 + 3.51e12T + 8.87e24T^{2} \)
89 \( 1 - 1.52e12T + 2.19e25T^{2} \)
97 \( 1 - 6.38e12T + 6.73e25T^{2} \)
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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

−10.35884860840724796602911243896, −9.671372214414533517424372558123, −7.996659318773873018612953633118, −7.03782108604176151422021411269, −5.83994681373315944118532286456, −5.28942808030976874057227073080, −4.04377698625612992075863577703, −3.07116404955206358086786443374, −2.35797562519475519258351487827, −1.18833937993312865135360979004, 1.18833937993312865135360979004, 2.35797562519475519258351487827, 3.07116404955206358086786443374, 4.04377698625612992075863577703, 5.28942808030976874057227073080, 5.83994681373315944118532286456, 7.03782108604176151422021411269, 7.996659318773873018612953633118, 9.671372214414533517424372558123, 10.35884860840724796602911243896

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