Properties

Label 2-177-1.1-c9-0-51
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  − 24.8·2-s + 81·3-s + 106.·4-s − 222.·5-s − 2.01e3·6-s − 2.49e3·7-s + 1.00e4·8-s + 6.56e3·9-s + 5.52e3·10-s − 1.77e4·11-s + 8.65e3·12-s − 1.22e4·13-s + 6.20e4·14-s − 1.79e4·15-s − 3.05e5·16-s − 1.70e5·17-s − 1.63e5·18-s + 8.00e5·19-s − 2.37e4·20-s − 2.02e5·21-s + 4.41e5·22-s − 4.41e4·23-s + 8.16e5·24-s − 1.90e6·25-s + 3.04e5·26-s + 5.31e5·27-s − 2.66e5·28-s + ⋯
L(s)  = 1  − 1.09·2-s + 0.577·3-s + 0.208·4-s − 0.158·5-s − 0.634·6-s − 0.392·7-s + 0.870·8-s + 0.333·9-s + 0.174·10-s − 0.365·11-s + 0.120·12-s − 0.118·13-s + 0.431·14-s − 0.0917·15-s − 1.16·16-s − 0.495·17-s − 0.366·18-s + 1.40·19-s − 0.0331·20-s − 0.226·21-s + 0.401·22-s − 0.0328·23-s + 0.502·24-s − 0.974·25-s + 0.130·26-s + 0.192·27-s − 0.0819·28-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 \( 1 + 24.8T + 512T^{2} \)
5 \( 1 + 222.T + 1.95e6T^{2} \)
7 \( 1 + 2.49e3T + 4.03e7T^{2} \)
11 \( 1 + 1.77e4T + 2.35e9T^{2} \)
13 \( 1 + 1.22e4T + 1.06e10T^{2} \)
17 \( 1 + 1.70e5T + 1.18e11T^{2} \)
19 \( 1 - 8.00e5T + 3.22e11T^{2} \)
23 \( 1 + 4.41e4T + 1.80e12T^{2} \)
29 \( 1 - 2.97e6T + 1.45e13T^{2} \)
31 \( 1 - 3.78e6T + 2.64e13T^{2} \)
37 \( 1 + 1.62e7T + 1.29e14T^{2} \)
41 \( 1 - 4.70e6T + 3.27e14T^{2} \)
43 \( 1 + 2.93e6T + 5.02e14T^{2} \)
47 \( 1 - 5.72e7T + 1.11e15T^{2} \)
53 \( 1 + 7.27e7T + 3.29e15T^{2} \)
61 \( 1 - 1.75e8T + 1.16e16T^{2} \)
67 \( 1 - 2.04e8T + 2.72e16T^{2} \)
71 \( 1 - 4.00e8T + 4.58e16T^{2} \)
73 \( 1 + 2.17e8T + 5.88e16T^{2} \)
79 \( 1 - 3.33e8T + 1.19e17T^{2} \)
83 \( 1 + 3.31e8T + 1.86e17T^{2} \)
89 \( 1 + 9.11e8T + 3.50e17T^{2} \)
97 \( 1 + 4.51e8T + 7.60e17T^{2} \)
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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

−10.09428116919793837831042812997, −9.529224295218802845721851585024, −8.521109836101561487922125778076, −7.75086302099210217880679168661, −6.79509215539231640411855051222, −5.13483372446712665013345153188, −3.82034583886528960307533054547, −2.48555168464690775593154751169, −1.16943632805176769134895926694, 0, 1.16943632805176769134895926694, 2.48555168464690775593154751169, 3.82034583886528960307533054547, 5.13483372446712665013345153188, 6.79509215539231640411855051222, 7.75086302099210217880679168661, 8.521109836101561487922125778076, 9.529224295218802845721851585024, 10.09428116919793837831042812997

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