Properties

Label 2-177-1.1-c9-0-4
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  − 12.8·2-s + 81·3-s − 346.·4-s − 288.·5-s − 1.04e3·6-s − 1.11e4·7-s + 1.10e4·8-s + 6.56e3·9-s + 3.70e3·10-s − 5.35e3·11-s − 2.81e4·12-s − 1.19e4·13-s + 1.43e5·14-s − 2.33e4·15-s + 3.58e4·16-s − 2.53e5·17-s − 8.42e4·18-s − 3.68e5·19-s + 9.99e4·20-s − 9.03e5·21-s + 6.87e4·22-s − 1.43e6·23-s + 8.93e5·24-s − 1.87e6·25-s + 1.53e5·26-s + 5.31e5·27-s + 3.87e6·28-s + ⋯
L(s)  = 1  − 0.567·2-s + 0.577·3-s − 0.677·4-s − 0.206·5-s − 0.327·6-s − 1.75·7-s + 0.952·8-s + 0.333·9-s + 0.117·10-s − 0.110·11-s − 0.391·12-s − 0.116·13-s + 0.997·14-s − 0.119·15-s + 0.136·16-s − 0.735·17-s − 0.189·18-s − 0.648·19-s + 0.139·20-s − 1.01·21-s + 0.0626·22-s − 1.06·23-s + 0.549·24-s − 0.957·25-s + 0.0659·26-s + 0.192·27-s + 1.19·28-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

\(L(5)\) \(\approx\) \(0.3371057685\)
\(L(\frac12)\) \(\approx\) \(0.3371057685\)
\(L(\frac{11}{2})\) 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 \( 1 + 12.8T + 512T^{2} \)
5 \( 1 + 288.T + 1.95e6T^{2} \)
7 \( 1 + 1.11e4T + 4.03e7T^{2} \)
11 \( 1 + 5.35e3T + 2.35e9T^{2} \)
13 \( 1 + 1.19e4T + 1.06e10T^{2} \)
17 \( 1 + 2.53e5T + 1.18e11T^{2} \)
19 \( 1 + 3.68e5T + 3.22e11T^{2} \)
23 \( 1 + 1.43e6T + 1.80e12T^{2} \)
29 \( 1 + 4.56e6T + 1.45e13T^{2} \)
31 \( 1 + 5.52e5T + 2.64e13T^{2} \)
37 \( 1 - 5.14e6T + 1.29e14T^{2} \)
41 \( 1 + 1.80e7T + 3.27e14T^{2} \)
43 \( 1 + 1.18e7T + 5.02e14T^{2} \)
47 \( 1 - 3.30e7T + 1.11e15T^{2} \)
53 \( 1 + 4.28e7T + 3.29e15T^{2} \)
61 \( 1 - 9.99e7T + 1.16e16T^{2} \)
67 \( 1 + 1.80e8T + 2.72e16T^{2} \)
71 \( 1 + 2.83e8T + 4.58e16T^{2} \)
73 \( 1 + 1.11e8T + 5.88e16T^{2} \)
79 \( 1 - 4.87e8T + 1.19e17T^{2} \)
83 \( 1 - 7.30e7T + 1.86e17T^{2} \)
89 \( 1 - 6.25e8T + 3.50e17T^{2} \)
97 \( 1 - 1.65e8T + 7.60e17T^{2} \)
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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

−10.58576884699469906785676099171, −9.742408346518968087680326705968, −9.157676422632193464233677170860, −8.159065239531232092533395545720, −7.10186969027604911573554729559, −5.92755998069749231608856788584, −4.27311709234337285828004765595, −3.43716657544367461228674620420, −2.02535742296626694591640008180, −0.28983805065045137118969105659, 0.28983805065045137118969105659, 2.02535742296626694591640008180, 3.43716657544367461228674620420, 4.27311709234337285828004765595, 5.92755998069749231608856788584, 7.10186969027604911573554729559, 8.159065239531232092533395545720, 9.157676422632193464233677170860, 9.742408346518968087680326705968, 10.58576884699469906785676099171

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