Properties

Label 2-177-1.1-c9-0-63
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  − 18.1·2-s + 81·3-s − 181.·4-s + 1.09e3·5-s − 1.47e3·6-s − 2.88e3·7-s + 1.26e4·8-s + 6.56e3·9-s − 1.99e4·10-s + 1.67e3·11-s − 1.47e4·12-s − 1.28e4·13-s + 5.24e4·14-s + 8.88e4·15-s − 1.35e5·16-s − 1.00e5·17-s − 1.19e5·18-s − 5.09e5·19-s − 1.99e5·20-s − 2.33e5·21-s − 3.04e4·22-s + 2.00e6·23-s + 1.02e6·24-s − 7.50e5·25-s + 2.33e5·26-s + 5.31e5·27-s + 5.24e5·28-s + ⋯
L(s)  = 1  − 0.803·2-s + 0.577·3-s − 0.355·4-s + 0.784·5-s − 0.463·6-s − 0.454·7-s + 1.08·8-s + 0.333·9-s − 0.630·10-s + 0.0345·11-s − 0.205·12-s − 0.124·13-s + 0.364·14-s + 0.452·15-s − 0.518·16-s − 0.292·17-s − 0.267·18-s − 0.896·19-s − 0.278·20-s − 0.262·21-s − 0.0277·22-s + 1.49·23-s + 0.628·24-s − 0.384·25-s + 0.100·26-s + 0.192·27-s + 0.161·28-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 \( 1 + 18.1T + 512T^{2} \)
5 \( 1 - 1.09e3T + 1.95e6T^{2} \)
7 \( 1 + 2.88e3T + 4.03e7T^{2} \)
11 \( 1 - 1.67e3T + 2.35e9T^{2} \)
13 \( 1 + 1.28e4T + 1.06e10T^{2} \)
17 \( 1 + 1.00e5T + 1.18e11T^{2} \)
19 \( 1 + 5.09e5T + 3.22e11T^{2} \)
23 \( 1 - 2.00e6T + 1.80e12T^{2} \)
29 \( 1 - 1.00e6T + 1.45e13T^{2} \)
31 \( 1 + 9.60e6T + 2.64e13T^{2} \)
37 \( 1 - 1.57e7T + 1.29e14T^{2} \)
41 \( 1 - 1.77e7T + 3.27e14T^{2} \)
43 \( 1 + 3.57e7T + 5.02e14T^{2} \)
47 \( 1 + 1.42e5T + 1.11e15T^{2} \)
53 \( 1 - 6.97e7T + 3.29e15T^{2} \)
61 \( 1 - 7.35e6T + 1.16e16T^{2} \)
67 \( 1 + 1.33e8T + 2.72e16T^{2} \)
71 \( 1 - 2.69e8T + 4.58e16T^{2} \)
73 \( 1 - 2.14e8T + 5.88e16T^{2} \)
79 \( 1 + 6.11e8T + 1.19e17T^{2} \)
83 \( 1 - 4.77e8T + 1.86e17T^{2} \)
89 \( 1 + 7.71e8T + 3.50e17T^{2} \)
97 \( 1 - 1.38e9T + 7.60e17T^{2} \)
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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

−10.16567359769265209029679001725, −9.369664084830197404481589073395, −8.791968337547547623506009443640, −7.66172301312610357510563186988, −6.56606740318991232504455517282, −5.16973462088667256984890951653, −3.90481829752719876303416520200, −2.44828828667770646848868843969, −1.33288465611583035828799815055, 0, 1.33288465611583035828799815055, 2.44828828667770646848868843969, 3.90481829752719876303416520200, 5.16973462088667256984890951653, 6.56606740318991232504455517282, 7.66172301312610357510563186988, 8.791968337547547623506009443640, 9.369664084830197404481589073395, 10.16567359769265209029679001725

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