Properties

Label 2-177-1.1-c13-0-90
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 102.·2-s + 729·3-s + 2.28e3·4-s + 1.52e3·5-s − 7.46e4·6-s − 1.06e5·7-s + 6.04e5·8-s + 5.31e5·9-s − 1.56e5·10-s + 6.31e6·11-s + 1.66e6·12-s + 2.25e7·13-s + 1.08e7·14-s + 1.11e6·15-s − 8.05e7·16-s − 5.20e7·17-s − 5.43e7·18-s − 1.84e8·19-s + 3.48e6·20-s − 7.75e7·21-s − 6.45e8·22-s + 9.02e8·23-s + 4.40e8·24-s − 1.21e9·25-s − 2.31e9·26-s + 3.87e8·27-s − 2.42e8·28-s + ⋯
L(s)  = 1  − 1.13·2-s + 0.577·3-s + 0.278·4-s + 0.0437·5-s − 0.652·6-s − 0.341·7-s + 0.815·8-s + 0.333·9-s − 0.0494·10-s + 1.07·11-s + 0.160·12-s + 1.29·13-s + 0.386·14-s + 0.0252·15-s − 1.20·16-s − 0.522·17-s − 0.376·18-s − 0.900·19-s + 0.0121·20-s − 0.197·21-s − 1.21·22-s + 1.27·23-s + 0.470·24-s − 0.998·25-s − 1.46·26-s + 0.192·27-s − 0.0952·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 729T \)
59 \( 1 - 4.21e10T \)
good2 \( 1 + 102.T + 8.19e3T^{2} \)
5 \( 1 - 1.52e3T + 1.22e9T^{2} \)
7 \( 1 + 1.06e5T + 9.68e10T^{2} \)
11 \( 1 - 6.31e6T + 3.45e13T^{2} \)
13 \( 1 - 2.25e7T + 3.02e14T^{2} \)
17 \( 1 + 5.20e7T + 9.90e15T^{2} \)
19 \( 1 + 1.84e8T + 4.20e16T^{2} \)
23 \( 1 - 9.02e8T + 5.04e17T^{2} \)
29 \( 1 + 6.01e9T + 1.02e19T^{2} \)
31 \( 1 - 4.64e8T + 2.44e19T^{2} \)
37 \( 1 + 3.80e9T + 2.43e20T^{2} \)
41 \( 1 - 4.44e10T + 9.25e20T^{2} \)
43 \( 1 + 2.92e10T + 1.71e21T^{2} \)
47 \( 1 - 6.84e10T + 5.46e21T^{2} \)
53 \( 1 + 2.50e11T + 2.60e22T^{2} \)
61 \( 1 - 1.92e9T + 1.61e23T^{2} \)
67 \( 1 + 4.84e11T + 5.48e23T^{2} \)
71 \( 1 + 7.15e11T + 1.16e24T^{2} \)
73 \( 1 + 1.30e12T + 1.67e24T^{2} \)
79 \( 1 - 1.45e12T + 4.66e24T^{2} \)
83 \( 1 - 5.72e12T + 8.87e24T^{2} \)
89 \( 1 + 6.89e12T + 2.19e25T^{2} \)
97 \( 1 + 2.72e12T + 6.73e25T^{2} \)
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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

−9.369239833113735550663184685752, −9.078074838594896973010346825270, −8.135553354372882063308855621240, −7.09029937265496067982627378912, −6.09113852037734557570770425661, −4.37737939581224482585320294499, −3.51395342920977675003721139553, −1.96854853981238126689046667548, −1.17813766934911134428685786230, 0, 1.17813766934911134428685786230, 1.96854853981238126689046667548, 3.51395342920977675003721139553, 4.37737939581224482585320294499, 6.09113852037734557570770425661, 7.09029937265496067982627378912, 8.135553354372882063308855621240, 9.078074838594896973010346825270, 9.369239833113735550663184685752

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