Properties

Label 2-177-1.1-c7-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.8·2-s − 27·3-s + 63.2·4-s − 149.·5-s + 373.·6-s + 14.2·7-s + 895.·8-s + 729·9-s + 2.06e3·10-s + 531.·11-s − 1.70e3·12-s + 4.16e3·13-s − 197.·14-s + 4.03e3·15-s − 2.04e4·16-s + 4.56e3·17-s − 1.00e4·18-s + 3.18e4·19-s − 9.45e3·20-s − 385.·21-s − 7.35e3·22-s − 5.60e4·23-s − 2.41e4·24-s − 5.57e4·25-s − 5.76e4·26-s − 1.96e4·27-s + 903.·28-s + ⋯
L(s)  = 1  − 1.22·2-s − 0.577·3-s + 0.494·4-s − 0.534·5-s + 0.705·6-s + 0.0157·7-s + 0.618·8-s + 0.333·9-s + 0.653·10-s + 0.120·11-s − 0.285·12-s + 0.526·13-s − 0.0192·14-s + 0.308·15-s − 1.24·16-s + 0.225·17-s − 0.407·18-s + 1.06·19-s − 0.264·20-s − 0.00908·21-s − 0.147·22-s − 0.960·23-s − 0.356·24-s − 0.714·25-s − 0.643·26-s − 0.192·27-s + 0.00777·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.5520223296\)
\(L(\frac12)\) \(\approx\) \(0.5520223296\)
\(L(\frac{9}{2})\) 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 \( 1 + 13.8T + 128T^{2} \)
5 \( 1 + 149.T + 7.81e4T^{2} \)
7 \( 1 - 14.2T + 8.23e5T^{2} \)
11 \( 1 - 531.T + 1.94e7T^{2} \)
13 \( 1 - 4.16e3T + 6.27e7T^{2} \)
17 \( 1 - 4.56e3T + 4.10e8T^{2} \)
19 \( 1 - 3.18e4T + 8.93e8T^{2} \)
23 \( 1 + 5.60e4T + 3.40e9T^{2} \)
29 \( 1 - 5.20e4T + 1.72e10T^{2} \)
31 \( 1 + 4.40e4T + 2.75e10T^{2} \)
37 \( 1 - 1.13e4T + 9.49e10T^{2} \)
41 \( 1 + 4.55e5T + 1.94e11T^{2} \)
43 \( 1 - 1.77e5T + 2.71e11T^{2} \)
47 \( 1 + 1.27e6T + 5.06e11T^{2} \)
53 \( 1 - 2.74e5T + 1.17e12T^{2} \)
61 \( 1 + 1.52e5T + 3.14e12T^{2} \)
67 \( 1 - 2.06e6T + 6.06e12T^{2} \)
71 \( 1 + 1.79e6T + 9.09e12T^{2} \)
73 \( 1 - 3.32e6T + 1.10e13T^{2} \)
79 \( 1 - 6.43e6T + 1.92e13T^{2} \)
83 \( 1 + 1.15e6T + 2.71e13T^{2} \)
89 \( 1 - 3.69e6T + 4.42e13T^{2} \)
97 \( 1 - 6.33e6T + 8.07e13T^{2} \)
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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

−11.28341085191120846664776020940, −10.23152997692028534914406648071, −9.482682211436727287208221709578, −8.286444338829260981001614593107, −7.57754871875625220506143784937, −6.39618270354167257928933158627, −4.98552690989814575959950772411, −3.66551492664650442096727457412, −1.67387161851102881113617326463, −0.52393267577116903636853752988, 0.52393267577116903636853752988, 1.67387161851102881113617326463, 3.66551492664650442096727457412, 4.98552690989814575959950772411, 6.39618270354167257928933158627, 7.57754871875625220506143784937, 8.286444338829260981001614593107, 9.482682211436727287208221709578, 10.23152997692028534914406648071, 11.28341085191120846664776020940

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