Properties

Label 2-177-1.1-c3-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.68·2-s + 3·3-s − 5.17·4-s − 9.78·5-s + 5.04·6-s + 6.87·7-s − 22.1·8-s + 9·9-s − 16.4·10-s − 37.8·11-s − 15.5·12-s − 11.3·13-s + 11.5·14-s − 29.3·15-s + 4.12·16-s − 60.3·17-s + 15.1·18-s − 92.5·19-s + 50.6·20-s + 20.6·21-s − 63.6·22-s − 183.·23-s − 66.4·24-s − 29.2·25-s − 19.0·26-s + 27·27-s − 35.5·28-s + ⋯
L(s)  = 1  + 0.594·2-s + 0.577·3-s − 0.646·4-s − 0.875·5-s + 0.343·6-s + 0.371·7-s − 0.978·8-s + 0.333·9-s − 0.520·10-s − 1.03·11-s − 0.373·12-s − 0.241·13-s + 0.220·14-s − 0.505·15-s + 0.0643·16-s − 0.860·17-s + 0.198·18-s − 1.11·19-s + 0.565·20-s + 0.214·21-s − 0.616·22-s − 1.66·23-s − 0.565·24-s − 0.233·25-s − 0.143·26-s + 0.192·27-s − 0.239·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.68T + 8T^{2} \)
5 \( 1 + 9.78T + 125T^{2} \)
7 \( 1 - 6.87T + 343T^{2} \)
11 \( 1 + 37.8T + 1.33e3T^{2} \)
13 \( 1 + 11.3T + 2.19e3T^{2} \)
17 \( 1 + 60.3T + 4.91e3T^{2} \)
19 \( 1 + 92.5T + 6.85e3T^{2} \)
23 \( 1 + 183.T + 1.21e4T^{2} \)
29 \( 1 - 265.T + 2.43e4T^{2} \)
31 \( 1 - 177.T + 2.97e4T^{2} \)
37 \( 1 + 70.0T + 5.06e4T^{2} \)
41 \( 1 - 208.T + 6.89e4T^{2} \)
43 \( 1 - 393.T + 7.95e4T^{2} \)
47 \( 1 + 134.T + 1.03e5T^{2} \)
53 \( 1 + 650.T + 1.48e5T^{2} \)
61 \( 1 - 22.5T + 2.26e5T^{2} \)
67 \( 1 + 209.T + 3.00e5T^{2} \)
71 \( 1 - 209.T + 3.57e5T^{2} \)
73 \( 1 - 865.T + 3.89e5T^{2} \)
79 \( 1 - 843.T + 4.93e5T^{2} \)
83 \( 1 + 845.T + 5.71e5T^{2} \)
89 \( 1 + 974.T + 7.04e5T^{2} \)
97 \( 1 - 338.T + 9.12e5T^{2} \)
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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

−12.06598782932537296473686986944, −10.79571650174012353228420527526, −9.676630822271598603371694445488, −8.345986232648966630955518299233, −7.974963296889829424368042041371, −6.30817627965155909779523464869, −4.75827180136345138551470840975, −4.05913429019322440350374346581, −2.59330988709362911747075112847, 0, 2.59330988709362911747075112847, 4.05913429019322440350374346581, 4.75827180136345138551470840975, 6.30817627965155909779523464869, 7.974963296889829424368042041371, 8.345986232648966630955518299233, 9.676630822271598603371694445488, 10.79571650174012353228420527526, 12.06598782932537296473686986944

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