Properties

Label 2-177-1.1-c13-0-71
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  − 126.·2-s + 729·3-s + 7.88e3·4-s + 5.70e4·5-s − 9.24e4·6-s + 5.43e5·7-s + 3.88e4·8-s + 5.31e5·9-s − 7.23e6·10-s + 4.27e6·11-s + 5.74e6·12-s + 1.89e7·13-s − 6.89e7·14-s + 4.16e7·15-s − 6.95e7·16-s + 6.68e7·17-s − 6.73e7·18-s − 1.27e8·19-s + 4.50e8·20-s + 3.96e8·21-s − 5.41e8·22-s − 1.18e9·23-s + 2.83e7·24-s + 2.03e9·25-s − 2.39e9·26-s + 3.87e8·27-s + 4.28e9·28-s + ⋯
L(s)  = 1  − 1.40·2-s + 0.577·3-s + 0.962·4-s + 1.63·5-s − 0.808·6-s + 1.74·7-s + 0.0523·8-s + 0.333·9-s − 2.28·10-s + 0.727·11-s + 0.555·12-s + 1.08·13-s − 2.44·14-s + 0.943·15-s − 1.03·16-s + 0.671·17-s − 0.466·18-s − 0.622·19-s + 1.57·20-s + 1.00·21-s − 1.01·22-s − 1.66·23-s + 0.0302·24-s + 1.66·25-s − 1.52·26-s + 0.192·27-s + 1.68·28-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

\(L(7)\) \(\approx\) \(3.090035053\)
\(L(\frac12)\) \(\approx\) \(3.090035053\)
\(L(\frac{15}{2})\) 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 \( 1 + 126.T + 8.19e3T^{2} \)
5 \( 1 - 5.70e4T + 1.22e9T^{2} \)
7 \( 1 - 5.43e5T + 9.68e10T^{2} \)
11 \( 1 - 4.27e6T + 3.45e13T^{2} \)
13 \( 1 - 1.89e7T + 3.02e14T^{2} \)
17 \( 1 - 6.68e7T + 9.90e15T^{2} \)
19 \( 1 + 1.27e8T + 4.20e16T^{2} \)
23 \( 1 + 1.18e9T + 5.04e17T^{2} \)
29 \( 1 + 4.39e8T + 1.02e19T^{2} \)
31 \( 1 + 4.50e8T + 2.44e19T^{2} \)
37 \( 1 + 1.47e10T + 2.43e20T^{2} \)
41 \( 1 + 1.44e10T + 9.25e20T^{2} \)
43 \( 1 + 6.17e9T + 1.71e21T^{2} \)
47 \( 1 - 1.00e11T + 5.46e21T^{2} \)
53 \( 1 - 3.49e10T + 2.60e22T^{2} \)
61 \( 1 - 4.06e11T + 1.61e23T^{2} \)
67 \( 1 - 1.91e11T + 5.48e23T^{2} \)
71 \( 1 + 1.26e12T + 1.16e24T^{2} \)
73 \( 1 - 7.26e11T + 1.67e24T^{2} \)
79 \( 1 - 1.13e12T + 4.66e24T^{2} \)
83 \( 1 - 4.87e12T + 8.87e24T^{2} \)
89 \( 1 - 3.48e12T + 2.19e25T^{2} \)
97 \( 1 - 1.18e13T + 6.73e25T^{2} \)
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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

−10.18831222435749603889158007404, −9.179388754180882726544812738529, −8.550693673435019528185582554668, −7.80703402676879289147410474176, −6.51010402280784872326788818603, −5.39508895424377866156984415362, −4.05781205908335944967018382887, −2.13236738092624435819000522906, −1.70349210208603195626200585982, −1.00119896492773795794032368422, 1.00119896492773795794032368422, 1.70349210208603195626200585982, 2.13236738092624435819000522906, 4.05781205908335944967018382887, 5.39508895424377866156984415362, 6.51010402280784872326788818603, 7.80703402676879289147410474176, 8.550693673435019528185582554668, 9.179388754180882726544812738529, 10.18831222435749603889158007404

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