Properties

Label 2-177-1.1-c9-0-27
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 28.8·2-s − 81·3-s + 321.·4-s − 2.14e3·5-s + 2.33e3·6-s + 2.49e3·7-s + 5.49e3·8-s + 6.56e3·9-s + 6.19e4·10-s − 2.48e4·11-s − 2.60e4·12-s − 1.27e5·13-s − 7.20e4·14-s + 1.73e5·15-s − 3.23e5·16-s − 4.24e5·17-s − 1.89e5·18-s − 3.08e5·19-s − 6.90e5·20-s − 2.02e5·21-s + 7.18e5·22-s + 1.81e6·23-s − 4.44e5·24-s + 2.64e6·25-s + 3.68e6·26-s − 5.31e5·27-s + 8.03e5·28-s + ⋯
L(s)  = 1  − 1.27·2-s − 0.577·3-s + 0.628·4-s − 1.53·5-s + 0.736·6-s + 0.392·7-s + 0.474·8-s + 0.333·9-s + 1.95·10-s − 0.512·11-s − 0.362·12-s − 1.23·13-s − 0.501·14-s + 0.886·15-s − 1.23·16-s − 1.23·17-s − 0.425·18-s − 0.543·19-s − 0.964·20-s − 0.226·21-s + 0.653·22-s + 1.35·23-s − 0.273·24-s + 1.35·25-s + 1.57·26-s − 0.192·27-s + 0.246·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 + 28.8T + 512T^{2} \)
5 \( 1 + 2.14e3T + 1.95e6T^{2} \)
7 \( 1 - 2.49e3T + 4.03e7T^{2} \)
11 \( 1 + 2.48e4T + 2.35e9T^{2} \)
13 \( 1 + 1.27e5T + 1.06e10T^{2} \)
17 \( 1 + 4.24e5T + 1.18e11T^{2} \)
19 \( 1 + 3.08e5T + 3.22e11T^{2} \)
23 \( 1 - 1.81e6T + 1.80e12T^{2} \)
29 \( 1 - 3.84e6T + 1.45e13T^{2} \)
31 \( 1 + 9.38e6T + 2.64e13T^{2} \)
37 \( 1 - 4.96e6T + 1.29e14T^{2} \)
41 \( 1 - 1.21e7T + 3.27e14T^{2} \)
43 \( 1 - 2.79e7T + 5.02e14T^{2} \)
47 \( 1 - 1.69e7T + 1.11e15T^{2} \)
53 \( 1 - 8.66e7T + 3.29e15T^{2} \)
61 \( 1 - 7.59e7T + 1.16e16T^{2} \)
67 \( 1 + 1.33e8T + 2.72e16T^{2} \)
71 \( 1 - 1.85e8T + 4.58e16T^{2} \)
73 \( 1 - 1.90e8T + 5.88e16T^{2} \)
79 \( 1 - 3.35e8T + 1.19e17T^{2} \)
83 \( 1 - 2.69e8T + 1.86e17T^{2} \)
89 \( 1 + 6.99e8T + 3.50e17T^{2} \)
97 \( 1 + 1.03e9T + 7.60e17T^{2} \)
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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

−10.79423124825681637499300405471, −9.381598958144489337043845543594, −8.456342816956697861137593151497, −7.53056763982142647099429898546, −6.96251057551360552887861515094, −4.99114331996046620432235985665, −4.19421514974768286947684858475, −2.36309267469576617236419942022, −0.75510310353169709631143697861, 0, 0.75510310353169709631143697861, 2.36309267469576617236419942022, 4.19421514974768286947684858475, 4.99114331996046620432235985665, 6.96251057551360552887861515094, 7.53056763982142647099429898546, 8.456342816956697861137593151497, 9.381598958144489337043845543594, 10.79423124825681637499300405471

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