Properties

Label 2-177-1.1-c9-0-8
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  − 36.1·2-s − 81·3-s + 796.·4-s + 898.·5-s + 2.93e3·6-s − 4.67e3·7-s − 1.02e4·8-s + 6.56e3·9-s − 3.25e4·10-s − 5.83e4·11-s − 6.45e4·12-s + 5.73e4·13-s + 1.69e5·14-s − 7.28e4·15-s − 3.54e4·16-s − 5.10e5·17-s − 2.37e5·18-s − 1.68e5·19-s + 7.16e5·20-s + 3.78e5·21-s + 2.11e6·22-s + 2.17e6·23-s + 8.33e5·24-s − 1.14e6·25-s − 2.07e6·26-s − 5.31e5·27-s − 3.72e6·28-s + ⋯
L(s)  = 1  − 1.59·2-s − 0.577·3-s + 1.55·4-s + 0.643·5-s + 0.923·6-s − 0.735·7-s − 0.888·8-s + 0.333·9-s − 1.02·10-s − 1.20·11-s − 0.898·12-s + 0.556·13-s + 1.17·14-s − 0.371·15-s − 0.135·16-s − 1.48·17-s − 0.532·18-s − 0.296·19-s + 1.00·20-s + 0.424·21-s + 1.92·22-s + 1.62·23-s + 0.513·24-s − 0.586·25-s − 0.890·26-s − 0.192·27-s − 1.14·28-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

\(L(5)\) \(\approx\) \(0.3625857366\)
\(L(\frac12)\) \(\approx\) \(0.3625857366\)
\(L(\frac{11}{2})\) 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 \( 1 + 36.1T + 512T^{2} \)
5 \( 1 - 898.T + 1.95e6T^{2} \)
7 \( 1 + 4.67e3T + 4.03e7T^{2} \)
11 \( 1 + 5.83e4T + 2.35e9T^{2} \)
13 \( 1 - 5.73e4T + 1.06e10T^{2} \)
17 \( 1 + 5.10e5T + 1.18e11T^{2} \)
19 \( 1 + 1.68e5T + 3.22e11T^{2} \)
23 \( 1 - 2.17e6T + 1.80e12T^{2} \)
29 \( 1 - 2.44e5T + 1.45e13T^{2} \)
31 \( 1 - 6.46e6T + 2.64e13T^{2} \)
37 \( 1 + 4.04e6T + 1.29e14T^{2} \)
41 \( 1 + 8.50e6T + 3.27e14T^{2} \)
43 \( 1 + 1.02e7T + 5.02e14T^{2} \)
47 \( 1 + 3.48e7T + 1.11e15T^{2} \)
53 \( 1 + 5.20e7T + 3.29e15T^{2} \)
61 \( 1 - 2.99e7T + 1.16e16T^{2} \)
67 \( 1 + 2.13e8T + 2.72e16T^{2} \)
71 \( 1 + 1.52e8T + 4.58e16T^{2} \)
73 \( 1 - 1.83e8T + 5.88e16T^{2} \)
79 \( 1 - 7.72e7T + 1.19e17T^{2} \)
83 \( 1 + 1.73e8T + 1.86e17T^{2} \)
89 \( 1 + 6.72e8T + 3.50e17T^{2} \)
97 \( 1 - 1.37e9T + 7.60e17T^{2} \)
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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

−10.68771634111766829402772508998, −10.00602461457880856736627397106, −9.118131792425313576889325203054, −8.194870040475847878685088094799, −6.92859299217628609707420153182, −6.23484882877353193866083278128, −4.83046391494017891225801159487, −2.82020077117817166005402021709, −1.66424871030387010306484586483, −0.38586747602496350738196565114, 0.38586747602496350738196565114, 1.66424871030387010306484586483, 2.82020077117817166005402021709, 4.83046391494017891225801159487, 6.23484882877353193866083278128, 6.92859299217628609707420153182, 8.194870040475847878685088094799, 9.118131792425313576889325203054, 10.00602461457880856736627397106, 10.68771634111766829402772508998

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