Properties

Label 2-177-1.1-c3-0-14
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.61·2-s − 3·3-s + 13.3·4-s − 3.21·5-s − 13.8·6-s + 15.9·7-s + 24.5·8-s + 9·9-s − 14.8·10-s + 54.2·11-s − 39.9·12-s + 85.2·13-s + 73.8·14-s + 9.65·15-s + 6.85·16-s − 48.1·17-s + 41.5·18-s + 64.2·19-s − 42.8·20-s − 47.9·21-s + 250.·22-s − 191.·23-s − 73.6·24-s − 114.·25-s + 393.·26-s − 27·27-s + 212.·28-s + ⋯
L(s)  = 1  + 1.63·2-s − 0.577·3-s + 1.66·4-s − 0.287·5-s − 0.942·6-s + 0.863·7-s + 1.08·8-s + 0.333·9-s − 0.469·10-s + 1.48·11-s − 0.961·12-s + 1.81·13-s + 1.40·14-s + 0.166·15-s + 0.107·16-s − 0.686·17-s + 0.544·18-s + 0.775·19-s − 0.479·20-s − 0.498·21-s + 2.42·22-s − 1.73·23-s − 0.626·24-s − 0.917·25-s + 2.97·26-s − 0.192·27-s + 1.43·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.893780610\)
\(L(\frac12)\) \(\approx\) \(3.893780610\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
59 \( 1 + 59T \)
good2 \( 1 - 4.61T + 8T^{2} \)
5 \( 1 + 3.21T + 125T^{2} \)
7 \( 1 - 15.9T + 343T^{2} \)
11 \( 1 - 54.2T + 1.33e3T^{2} \)
13 \( 1 - 85.2T + 2.19e3T^{2} \)
17 \( 1 + 48.1T + 4.91e3T^{2} \)
19 \( 1 - 64.2T + 6.85e3T^{2} \)
23 \( 1 + 191.T + 1.21e4T^{2} \)
29 \( 1 + 15.0T + 2.43e4T^{2} \)
31 \( 1 + 209.T + 2.97e4T^{2} \)
37 \( 1 - 418.T + 5.06e4T^{2} \)
41 \( 1 - 226.T + 6.89e4T^{2} \)
43 \( 1 + 207.T + 7.95e4T^{2} \)
47 \( 1 - 330.T + 1.03e5T^{2} \)
53 \( 1 + 449.T + 1.48e5T^{2} \)
61 \( 1 + 393.T + 2.26e5T^{2} \)
67 \( 1 + 67.7T + 3.00e5T^{2} \)
71 \( 1 + 589.T + 3.57e5T^{2} \)
73 \( 1 + 229.T + 3.89e5T^{2} \)
79 \( 1 - 563.T + 4.93e5T^{2} \)
83 \( 1 - 1.17e3T + 5.71e5T^{2} \)
89 \( 1 + 1.33e3T + 7.04e5T^{2} \)
97 \( 1 - 1.36e3T + 9.12e5T^{2} \)
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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

−12.14784755588953020622399273541, −11.47413776618721293590312900219, −11.01997975403699035975123182291, −9.221957578574429908519148552599, −7.79489349199683578610563066916, −6.37496078470151327749017481618, −5.80362385700175039609418471032, −4.34699508255869451716745833988, −3.76182584547269052236889615909, −1.62018724410396158492005331826, 1.62018724410396158492005331826, 3.76182584547269052236889615909, 4.34699508255869451716745833988, 5.80362385700175039609418471032, 6.37496078470151327749017481618, 7.79489349199683578610563066916, 9.221957578574429908519148552599, 11.01997975403699035975123182291, 11.47413776618721293590312900219, 12.14784755588953020622399273541

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