Properties

Label 2-177-1.1-c7-0-55
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 12.1·2-s + 27·3-s + 19.3·4-s + 236.·5-s − 327.·6-s + 1.42e3·7-s + 1.31e3·8-s + 729·9-s − 2.86e3·10-s − 5.47e3·11-s + 522.·12-s − 8.45e3·13-s − 1.73e4·14-s + 6.38e3·15-s − 1.84e4·16-s − 6.08e3·17-s − 8.84e3·18-s − 1.29e4·19-s + 4.57e3·20-s + 3.85e4·21-s + 6.64e4·22-s − 5.39e4·23-s + 3.56e4·24-s − 2.22e4·25-s + 1.02e5·26-s + 1.96e4·27-s + 2.76e4·28-s + ⋯
L(s)  = 1  − 1.07·2-s + 0.577·3-s + 0.151·4-s + 0.845·5-s − 0.619·6-s + 1.57·7-s + 0.910·8-s + 0.333·9-s − 0.907·10-s − 1.23·11-s + 0.0873·12-s − 1.06·13-s − 1.68·14-s + 0.488·15-s − 1.12·16-s − 0.300·17-s − 0.357·18-s − 0.433·19-s + 0.127·20-s + 0.907·21-s + 1.33·22-s − 0.925·23-s + 0.525·24-s − 0.285·25-s + 1.14·26-s + 0.192·27-s + 0.237·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 27T \)
59 \( 1 + 2.05e5T \)
good2 \( 1 + 12.1T + 128T^{2} \)
5 \( 1 - 236.T + 7.81e4T^{2} \)
7 \( 1 - 1.42e3T + 8.23e5T^{2} \)
11 \( 1 + 5.47e3T + 1.94e7T^{2} \)
13 \( 1 + 8.45e3T + 6.27e7T^{2} \)
17 \( 1 + 6.08e3T + 4.10e8T^{2} \)
19 \( 1 + 1.29e4T + 8.93e8T^{2} \)
23 \( 1 + 5.39e4T + 3.40e9T^{2} \)
29 \( 1 + 1.43e4T + 1.72e10T^{2} \)
31 \( 1 + 4.84e4T + 2.75e10T^{2} \)
37 \( 1 - 8.28e4T + 9.49e10T^{2} \)
41 \( 1 + 7.82e5T + 1.94e11T^{2} \)
43 \( 1 + 3.69e5T + 2.71e11T^{2} \)
47 \( 1 - 3.68e5T + 5.06e11T^{2} \)
53 \( 1 - 8.36e5T + 1.17e12T^{2} \)
61 \( 1 - 3.73e4T + 3.14e12T^{2} \)
67 \( 1 - 2.64e6T + 6.06e12T^{2} \)
71 \( 1 + 2.03e6T + 9.09e12T^{2} \)
73 \( 1 + 3.64e6T + 1.10e13T^{2} \)
79 \( 1 + 7.63e6T + 1.92e13T^{2} \)
83 \( 1 + 6.69e6T + 2.71e13T^{2} \)
89 \( 1 - 7.17e5T + 4.42e13T^{2} \)
97 \( 1 - 1.06e7T + 8.07e13T^{2} \)
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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

−10.39966294614909563839860067099, −9.945874034494061471558135418044, −8.735681492055786641154260981859, −8.059590367850017130745347951633, −7.27592121809788550627796916453, −5.39064571960200604360136255650, −4.48496486458400643315626290780, −2.30854229768775414944186637012, −1.65972904065695586245002404394, 0, 1.65972904065695586245002404394, 2.30854229768775414944186637012, 4.48496486458400643315626290780, 5.39064571960200604360136255650, 7.27592121809788550627796916453, 8.059590367850017130745347951633, 8.735681492055786641154260981859, 9.945874034494061471558135418044, 10.39966294614909563839860067099

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