Properties

Label 2-177-1.1-c7-0-54
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.413·2-s + 27·3-s − 127.·4-s + 231.·5-s + 11.1·6-s + 302.·7-s − 105.·8-s + 729·9-s + 95.4·10-s + 1.28e3·11-s − 3.45e3·12-s − 1.48e4·13-s + 124.·14-s + 6.24e3·15-s + 1.63e4·16-s − 2.11e4·17-s + 301.·18-s − 1.75e3·19-s − 2.95e4·20-s + 8.15e3·21-s + 529.·22-s + 9.19e4·23-s − 2.85e3·24-s − 2.46e4·25-s − 6.13e3·26-s + 1.96e4·27-s − 3.86e4·28-s + ⋯
L(s)  = 1  + 0.0365·2-s + 0.577·3-s − 0.998·4-s + 0.826·5-s + 0.0210·6-s + 0.332·7-s − 0.0729·8-s + 0.333·9-s + 0.0301·10-s + 0.290·11-s − 0.576·12-s − 1.87·13-s + 0.0121·14-s + 0.477·15-s + 0.996·16-s − 1.04·17-s + 0.0121·18-s − 0.0587·19-s − 0.825·20-s + 0.192·21-s + 0.0105·22-s + 1.57·23-s − 0.0421·24-s − 0.316·25-s − 0.0684·26-s + 0.192·27-s − 0.332·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 - 0.413T + 128T^{2} \)
5 \( 1 - 231.T + 7.81e4T^{2} \)
7 \( 1 - 302.T + 8.23e5T^{2} \)
11 \( 1 - 1.28e3T + 1.94e7T^{2} \)
13 \( 1 + 1.48e4T + 6.27e7T^{2} \)
17 \( 1 + 2.11e4T + 4.10e8T^{2} \)
19 \( 1 + 1.75e3T + 8.93e8T^{2} \)
23 \( 1 - 9.19e4T + 3.40e9T^{2} \)
29 \( 1 - 6.26e4T + 1.72e10T^{2} \)
31 \( 1 + 1.03e5T + 2.75e10T^{2} \)
37 \( 1 + 3.79e4T + 9.49e10T^{2} \)
41 \( 1 + 4.56e4T + 1.94e11T^{2} \)
43 \( 1 + 6.37e5T + 2.71e11T^{2} \)
47 \( 1 - 2.29e5T + 5.06e11T^{2} \)
53 \( 1 + 1.07e6T + 1.17e12T^{2} \)
61 \( 1 + 2.75e6T + 3.14e12T^{2} \)
67 \( 1 + 3.11e6T + 6.06e12T^{2} \)
71 \( 1 - 4.01e6T + 9.09e12T^{2} \)
73 \( 1 + 5.86e6T + 1.10e13T^{2} \)
79 \( 1 - 4.62e6T + 1.92e13T^{2} \)
83 \( 1 - 9.51e6T + 2.71e13T^{2} \)
89 \( 1 + 6.95e6T + 4.42e13T^{2} \)
97 \( 1 + 9.56e6T + 8.07e13T^{2} \)
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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

−10.64271726844215955011214434059, −9.551361429767915343719832078916, −9.143240513961974742367535352590, −7.955417210836021252709175000853, −6.77517416338945144958649724160, −5.22187264193679555664516205590, −4.48837522185806432630884796351, −2.91526019340627716514262009652, −1.65594856475175877654066826142, 0, 1.65594856475175877654066826142, 2.91526019340627716514262009652, 4.48837522185806432630884796351, 5.22187264193679555664516205590, 6.77517416338945144958649724160, 7.955417210836021252709175000853, 9.143240513961974742367535352590, 9.551361429767915343719832078916, 10.64271726844215955011214434059

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