Properties

Label 2-177-1.1-c7-0-25
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  − 19.9·2-s + 27·3-s + 270.·4-s + 505.·5-s − 538.·6-s + 222.·7-s − 2.83e3·8-s + 729·9-s − 1.00e4·10-s − 1.83e3·11-s + 7.29e3·12-s + 688.·13-s − 4.44e3·14-s + 1.36e4·15-s + 2.19e4·16-s + 2.84e3·17-s − 1.45e4·18-s + 3.44e4·19-s + 1.36e5·20-s + 6.02e3·21-s + 3.65e4·22-s − 3.27e3·23-s − 7.64e4·24-s + 1.77e5·25-s − 1.37e4·26-s + 1.96e4·27-s + 6.02e4·28-s + ⋯
L(s)  = 1  − 1.76·2-s + 0.577·3-s + 2.10·4-s + 1.80·5-s − 1.01·6-s + 0.245·7-s − 1.95·8-s + 0.333·9-s − 3.19·10-s − 0.414·11-s + 1.21·12-s + 0.0869·13-s − 0.433·14-s + 1.04·15-s + 1.34·16-s + 0.140·17-s − 0.587·18-s + 1.15·19-s + 3.81·20-s + 0.141·21-s + 0.731·22-s − 0.0560·23-s − 1.12·24-s + 2.27·25-s − 0.153·26-s + 0.192·27-s + 0.518·28-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

\(L(4)\) \(\approx\) \(1.808048211\)
\(L(\frac12)\) \(\approx\) \(1.808048211\)
\(L(\frac{9}{2})\) 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 \( 1 + 19.9T + 128T^{2} \)
5 \( 1 - 505.T + 7.81e4T^{2} \)
7 \( 1 - 222.T + 8.23e5T^{2} \)
11 \( 1 + 1.83e3T + 1.94e7T^{2} \)
13 \( 1 - 688.T + 6.27e7T^{2} \)
17 \( 1 - 2.84e3T + 4.10e8T^{2} \)
19 \( 1 - 3.44e4T + 8.93e8T^{2} \)
23 \( 1 + 3.27e3T + 3.40e9T^{2} \)
29 \( 1 - 1.28e5T + 1.72e10T^{2} \)
31 \( 1 + 1.65e4T + 2.75e10T^{2} \)
37 \( 1 + 4.24e5T + 9.49e10T^{2} \)
41 \( 1 + 1.25e4T + 1.94e11T^{2} \)
43 \( 1 - 3.38e5T + 2.71e11T^{2} \)
47 \( 1 - 5.60e5T + 5.06e11T^{2} \)
53 \( 1 - 2.04e6T + 1.17e12T^{2} \)
61 \( 1 + 2.42e6T + 3.14e12T^{2} \)
67 \( 1 + 2.63e6T + 6.06e12T^{2} \)
71 \( 1 - 2.62e6T + 9.09e12T^{2} \)
73 \( 1 - 3.21e6T + 1.10e13T^{2} \)
79 \( 1 - 3.42e6T + 1.92e13T^{2} \)
83 \( 1 - 1.20e6T + 2.71e13T^{2} \)
89 \( 1 + 2.50e6T + 4.42e13T^{2} \)
97 \( 1 - 8.73e6T + 8.07e13T^{2} \)
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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

−10.72508664057405078546983741220, −10.11177648484387002116321858416, −9.361496285965920665696383028938, −8.663239430345910027805960744076, −7.55432094986744426523274153567, −6.51871948903833711883788512223, −5.32463033752247726613683560332, −2.83445056989129761118412378835, −1.90648746126179472177499629214, −0.990596191157991163419761036133, 0.990596191157991163419761036133, 1.90648746126179472177499629214, 2.83445056989129761118412378835, 5.32463033752247726613683560332, 6.51871948903833711883788512223, 7.55432094986744426523274153567, 8.663239430345910027805960744076, 9.361496285965920665696383028938, 10.11177648484387002116321858416, 10.72508664057405078546983741220

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