Properties

Label 2-177-1.1-c9-0-36
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  − 39.1·2-s + 81·3-s + 1.01e3·4-s + 1.84e3·5-s − 3.16e3·6-s + 8.71e3·7-s − 1.97e4·8-s + 6.56e3·9-s − 7.21e4·10-s + 1.15e4·11-s + 8.23e4·12-s + 8.11e3·13-s − 3.40e5·14-s + 1.49e5·15-s + 2.51e5·16-s − 2.58e5·17-s − 2.56e5·18-s − 5.11e5·19-s + 1.87e6·20-s + 7.06e5·21-s − 4.53e5·22-s + 1.50e6·23-s − 1.59e6·24-s + 1.45e6·25-s − 3.17e5·26-s + 5.31e5·27-s + 8.86e6·28-s + ⋯
L(s)  = 1  − 1.72·2-s + 0.577·3-s + 1.98·4-s + 1.32·5-s − 0.997·6-s + 1.37·7-s − 1.70·8-s + 0.333·9-s − 2.28·10-s + 0.238·11-s + 1.14·12-s + 0.0787·13-s − 2.37·14-s + 0.762·15-s + 0.959·16-s − 0.751·17-s − 0.576·18-s − 0.900·19-s + 2.62·20-s + 0.792·21-s − 0.412·22-s + 1.12·23-s − 0.984·24-s + 0.743·25-s − 0.136·26-s + 0.192·27-s + 2.72·28-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

\(L(5)\) \(\approx\) \(2.042903166\)
\(L(\frac12)\) \(\approx\) \(2.042903166\)
\(L(\frac{11}{2})\) 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 \( 1 + 39.1T + 512T^{2} \)
5 \( 1 - 1.84e3T + 1.95e6T^{2} \)
7 \( 1 - 8.71e3T + 4.03e7T^{2} \)
11 \( 1 - 1.15e4T + 2.35e9T^{2} \)
13 \( 1 - 8.11e3T + 1.06e10T^{2} \)
17 \( 1 + 2.58e5T + 1.18e11T^{2} \)
19 \( 1 + 5.11e5T + 3.22e11T^{2} \)
23 \( 1 - 1.50e6T + 1.80e12T^{2} \)
29 \( 1 - 3.60e6T + 1.45e13T^{2} \)
31 \( 1 - 5.17e5T + 2.64e13T^{2} \)
37 \( 1 - 5.37e6T + 1.29e14T^{2} \)
41 \( 1 - 1.04e7T + 3.27e14T^{2} \)
43 \( 1 - 1.15e6T + 5.02e14T^{2} \)
47 \( 1 - 2.05e7T + 1.11e15T^{2} \)
53 \( 1 + 6.04e7T + 3.29e15T^{2} \)
61 \( 1 - 1.74e8T + 1.16e16T^{2} \)
67 \( 1 - 7.95e7T + 2.72e16T^{2} \)
71 \( 1 - 1.53e8T + 4.58e16T^{2} \)
73 \( 1 + 3.49e8T + 5.88e16T^{2} \)
79 \( 1 + 3.98e8T + 1.19e17T^{2} \)
83 \( 1 + 4.09e6T + 1.86e17T^{2} \)
89 \( 1 - 2.94e8T + 3.50e17T^{2} \)
97 \( 1 + 4.22e8T + 7.60e17T^{2} \)
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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

−10.69410201661977182530100731102, −9.815862787996613911786845341148, −8.894661277770774302736041756758, −8.386282078757159749321237859854, −7.24550518327418508703405923684, −6.20959173784272378650857970265, −4.68459223846610788841606420706, −2.51041637421155693376144076723, −1.81535654463232563533935239719, −0.946798812473760523455292331044, 0.946798812473760523455292331044, 1.81535654463232563533935239719, 2.51041637421155693376144076723, 4.68459223846610788841606420706, 6.20959173784272378650857970265, 7.24550518327418508703405923684, 8.386282078757159749321237859854, 8.894661277770774302736041756758, 9.815862787996613911786845341148, 10.69410201661977182530100731102

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