Properties

Label 2-177-1.1-c9-0-57
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 42.1·2-s − 81·3-s + 1.26e3·4-s + 1.72e3·5-s − 3.41e3·6-s − 4.21e3·7-s + 3.18e4·8-s + 6.56e3·9-s + 7.27e4·10-s + 8.17e4·11-s − 1.02e5·12-s − 4.73e4·13-s − 1.77e5·14-s − 1.39e5·15-s + 6.94e5·16-s − 6.03e4·17-s + 2.76e5·18-s + 4.88e5·19-s + 2.18e6·20-s + 3.41e5·21-s + 3.44e6·22-s − 6.64e5·23-s − 2.57e6·24-s + 1.02e6·25-s − 1.99e6·26-s − 5.31e5·27-s − 5.34e6·28-s + ⋯
L(s)  = 1  + 1.86·2-s − 0.577·3-s + 2.47·4-s + 1.23·5-s − 1.07·6-s − 0.663·7-s + 2.74·8-s + 0.333·9-s + 2.30·10-s + 1.68·11-s − 1.42·12-s − 0.459·13-s − 1.23·14-s − 0.712·15-s + 2.64·16-s − 0.175·17-s + 0.621·18-s + 0.859·19-s + 3.05·20-s + 0.383·21-s + 3.13·22-s − 0.495·23-s − 1.58·24-s + 0.523·25-s − 0.856·26-s − 0.192·27-s − 1.64·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(8.523369498\)
\(L(\frac12)\) \(\approx\) \(8.523369498\)
\(L(\frac{11}{2})\) 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 \( 1 - 42.1T + 512T^{2} \)
5 \( 1 - 1.72e3T + 1.95e6T^{2} \)
7 \( 1 + 4.21e3T + 4.03e7T^{2} \)
11 \( 1 - 8.17e4T + 2.35e9T^{2} \)
13 \( 1 + 4.73e4T + 1.06e10T^{2} \)
17 \( 1 + 6.03e4T + 1.18e11T^{2} \)
19 \( 1 - 4.88e5T + 3.22e11T^{2} \)
23 \( 1 + 6.64e5T + 1.80e12T^{2} \)
29 \( 1 + 5.04e5T + 1.45e13T^{2} \)
31 \( 1 - 2.59e6T + 2.64e13T^{2} \)
37 \( 1 - 1.60e7T + 1.29e14T^{2} \)
41 \( 1 + 2.57e6T + 3.27e14T^{2} \)
43 \( 1 + 1.94e7T + 5.02e14T^{2} \)
47 \( 1 - 4.00e7T + 1.11e15T^{2} \)
53 \( 1 - 8.87e7T + 3.29e15T^{2} \)
61 \( 1 + 6.53e7T + 1.16e16T^{2} \)
67 \( 1 + 2.63e8T + 2.72e16T^{2} \)
71 \( 1 - 6.50e7T + 4.58e16T^{2} \)
73 \( 1 - 2.39e8T + 5.88e16T^{2} \)
79 \( 1 + 3.02e8T + 1.19e17T^{2} \)
83 \( 1 - 2.55e8T + 1.86e17T^{2} \)
89 \( 1 + 4.80e7T + 3.50e17T^{2} \)
97 \( 1 + 1.83e8T + 7.60e17T^{2} \)
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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

−11.54448702678425386277384052084, −10.23747537167949332069384168780, −9.377881339594934260480253043782, −7.17201244237433697784775428637, −6.29325510496045971372403248203, −5.83269565968385529030897827549, −4.68372903775587009480055473514, −3.62315649699084667444914926605, −2.37250589135166338323347220297, −1.25164313540877634627582641624, 1.25164313540877634627582641624, 2.37250589135166338323347220297, 3.62315649699084667444914926605, 4.68372903775587009480055473514, 5.83269565968385529030897827549, 6.29325510496045971372403248203, 7.17201244237433697784775428637, 9.377881339594934260480253043782, 10.23747537167949332069384168780, 11.54448702678425386277384052084

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