Properties

Label 2-177-1.1-c7-0-48
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  + 20.9·2-s + 27·3-s + 310.·4-s − 149.·5-s + 565.·6-s + 1.20e3·7-s + 3.81e3·8-s + 729·9-s − 3.13e3·10-s + 3.09e3·11-s + 8.37e3·12-s − 9.76e3·13-s + 2.51e4·14-s − 4.04e3·15-s + 4.01e4·16-s + 774.·17-s + 1.52e4·18-s + 4.22e4·19-s − 4.64e4·20-s + 3.24e4·21-s + 6.46e4·22-s + 1.92e4·23-s + 1.02e5·24-s − 5.56e4·25-s − 2.04e5·26-s + 1.96e4·27-s + 3.72e5·28-s + ⋯
L(s)  = 1  + 1.85·2-s + 0.577·3-s + 2.42·4-s − 0.536·5-s + 1.06·6-s + 1.32·7-s + 2.63·8-s + 0.333·9-s − 0.992·10-s + 0.700·11-s + 1.39·12-s − 1.23·13-s + 2.44·14-s − 0.309·15-s + 2.44·16-s + 0.0382·17-s + 0.616·18-s + 1.41·19-s − 1.29·20-s + 0.764·21-s + 1.29·22-s + 0.330·23-s + 1.52·24-s − 0.712·25-s − 2.28·26-s + 0.192·27-s + 3.20·28-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

\(L(4)\) \(\approx\) \(9.280919207\)
\(L(\frac12)\) \(\approx\) \(9.280919207\)
\(L(\frac{9}{2})\) 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 \( 1 - 20.9T + 128T^{2} \)
5 \( 1 + 149.T + 7.81e4T^{2} \)
7 \( 1 - 1.20e3T + 8.23e5T^{2} \)
11 \( 1 - 3.09e3T + 1.94e7T^{2} \)
13 \( 1 + 9.76e3T + 6.27e7T^{2} \)
17 \( 1 - 774.T + 4.10e8T^{2} \)
19 \( 1 - 4.22e4T + 8.93e8T^{2} \)
23 \( 1 - 1.92e4T + 3.40e9T^{2} \)
29 \( 1 + 1.48e5T + 1.72e10T^{2} \)
31 \( 1 + 7.75e4T + 2.75e10T^{2} \)
37 \( 1 - 3.81e5T + 9.49e10T^{2} \)
41 \( 1 - 3.13e5T + 1.94e11T^{2} \)
43 \( 1 - 8.68e4T + 2.71e11T^{2} \)
47 \( 1 - 1.09e6T + 5.06e11T^{2} \)
53 \( 1 + 1.35e6T + 1.17e12T^{2} \)
61 \( 1 + 1.17e6T + 3.14e12T^{2} \)
67 \( 1 + 2.20e6T + 6.06e12T^{2} \)
71 \( 1 - 6.43e4T + 9.09e12T^{2} \)
73 \( 1 + 1.88e6T + 1.10e13T^{2} \)
79 \( 1 + 2.67e5T + 1.92e13T^{2} \)
83 \( 1 + 9.61e6T + 2.71e13T^{2} \)
89 \( 1 - 1.26e7T + 4.42e13T^{2} \)
97 \( 1 + 7.21e6T + 8.07e13T^{2} \)
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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

−11.70445324802665274900538704638, −11.01671184063806528454091203813, −9.427625287452834822017043328712, −7.71926399952838184465836539157, −7.31215515808210737225241999818, −5.69308759915987088068471124013, −4.70223645061993176857180524692, −3.90717711000770242602183088294, −2.68958756171618581095938690576, −1.51789753634137289784327516974, 1.51789753634137289784327516974, 2.68958756171618581095938690576, 3.90717711000770242602183088294, 4.70223645061993176857180524692, 5.69308759915987088068471124013, 7.31215515808210737225241999818, 7.71926399952838184465836539157, 9.427625287452834822017043328712, 11.01671184063806528454091203813, 11.70445324802665274900538704638

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