Properties

Label 2-177-1.1-c9-0-38
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  − 3.19·2-s + 81·3-s − 501.·4-s − 2.71e3·5-s − 258.·6-s − 1.84e3·7-s + 3.23e3·8-s + 6.56e3·9-s + 8.65e3·10-s + 1.41e4·11-s − 4.06e4·12-s + 1.30e5·13-s + 5.88e3·14-s − 2.19e5·15-s + 2.46e5·16-s − 5.50e5·17-s − 2.09e4·18-s − 7.66e4·19-s + 1.35e6·20-s − 1.49e5·21-s − 4.51e4·22-s + 1.37e6·23-s + 2.62e5·24-s + 5.39e6·25-s − 4.15e5·26-s + 5.31e5·27-s + 9.24e5·28-s + ⋯
L(s)  = 1  − 0.141·2-s + 0.577·3-s − 0.980·4-s − 1.93·5-s − 0.0814·6-s − 0.289·7-s + 0.279·8-s + 0.333·9-s + 0.273·10-s + 0.291·11-s − 0.565·12-s + 1.26·13-s + 0.0409·14-s − 1.11·15-s + 0.940·16-s − 1.59·17-s − 0.0470·18-s − 0.135·19-s + 1.90·20-s − 0.167·21-s − 0.0411·22-s + 1.02·23-s + 0.161·24-s + 2.76·25-s − 0.178·26-s + 0.192·27-s + 0.284·28-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 \( 1 + 3.19T + 512T^{2} \)
5 \( 1 + 2.71e3T + 1.95e6T^{2} \)
7 \( 1 + 1.84e3T + 4.03e7T^{2} \)
11 \( 1 - 1.41e4T + 2.35e9T^{2} \)
13 \( 1 - 1.30e5T + 1.06e10T^{2} \)
17 \( 1 + 5.50e5T + 1.18e11T^{2} \)
19 \( 1 + 7.66e4T + 3.22e11T^{2} \)
23 \( 1 - 1.37e6T + 1.80e12T^{2} \)
29 \( 1 - 1.57e6T + 1.45e13T^{2} \)
31 \( 1 - 2.58e6T + 2.64e13T^{2} \)
37 \( 1 - 3.46e5T + 1.29e14T^{2} \)
41 \( 1 + 2.25e7T + 3.27e14T^{2} \)
43 \( 1 - 3.52e7T + 5.02e14T^{2} \)
47 \( 1 - 3.28e7T + 1.11e15T^{2} \)
53 \( 1 + 5.90e7T + 3.29e15T^{2} \)
61 \( 1 + 1.93e7T + 1.16e16T^{2} \)
67 \( 1 + 2.82e8T + 2.72e16T^{2} \)
71 \( 1 - 6.18e7T + 4.58e16T^{2} \)
73 \( 1 - 3.00e8T + 5.88e16T^{2} \)
79 \( 1 + 2.20e8T + 1.19e17T^{2} \)
83 \( 1 + 2.37e8T + 1.86e17T^{2} \)
89 \( 1 + 6.63e8T + 3.50e17T^{2} \)
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

−10.62186218750776120907251976190, −9.009016344988153158911728631087, −8.640361265418336622770833380391, −7.71851371890640299568327043605, −6.60689449348110192784233072903, −4.65755972741660235010458754256, −3.99937585464075979548675727197, −3.12941480822212768366014390238, −1.03089943410378369572237681551, 0, 1.03089943410378369572237681551, 3.12941480822212768366014390238, 3.99937585464075979548675727197, 4.65755972741660235010458754256, 6.60689449348110192784233072903, 7.71851371890640299568327043605, 8.640361265418336622770833380391, 9.009016344988153158911728631087, 10.62186218750776120907251976190

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