Properties

Label 2-177-1.1-c13-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 90.0·2-s + 729·3-s − 84.2·4-s − 4.01e4·5-s + 6.56e4·6-s − 3.37e5·7-s − 7.45e5·8-s + 5.31e5·9-s − 3.61e6·10-s − 7.78e6·11-s − 6.14e4·12-s + 1.67e6·13-s − 3.04e7·14-s − 2.92e7·15-s − 6.64e7·16-s − 1.97e8·17-s + 4.78e7·18-s + 1.04e8·19-s + 3.38e6·20-s − 2.46e8·21-s − 7.00e8·22-s − 6.39e8·23-s − 5.43e8·24-s + 3.94e8·25-s + 1.50e8·26-s + 3.87e8·27-s + 2.84e7·28-s + ⋯
L(s)  = 1  + 0.994·2-s + 0.577·3-s − 0.0102·4-s − 1.15·5-s + 0.574·6-s − 1.08·7-s − 1.00·8-s + 0.333·9-s − 1.14·10-s − 1.32·11-s − 0.00593·12-s + 0.0959·13-s − 1.08·14-s − 0.664·15-s − 0.989·16-s − 1.98·17-s + 0.331·18-s + 0.509·19-s + 0.0118·20-s − 0.626·21-s − 1.31·22-s − 0.901·23-s − 0.580·24-s + 0.322·25-s + 0.0954·26-s + 0.192·27-s + 0.0111·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(0.1528438884\)
\(L(\frac12)\) \(\approx\) \(0.1528438884\)
\(L(\frac{15}{2})\) 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 \( 1 - 90.0T + 8.19e3T^{2} \)
5 \( 1 + 4.01e4T + 1.22e9T^{2} \)
7 \( 1 + 3.37e5T + 9.68e10T^{2} \)
11 \( 1 + 7.78e6T + 3.45e13T^{2} \)
13 \( 1 - 1.67e6T + 3.02e14T^{2} \)
17 \( 1 + 1.97e8T + 9.90e15T^{2} \)
19 \( 1 - 1.04e8T + 4.20e16T^{2} \)
23 \( 1 + 6.39e8T + 5.04e17T^{2} \)
29 \( 1 + 2.05e9T + 1.02e19T^{2} \)
31 \( 1 - 2.85e9T + 2.44e19T^{2} \)
37 \( 1 + 2.15e10T + 2.43e20T^{2} \)
41 \( 1 - 1.18e9T + 9.25e20T^{2} \)
43 \( 1 + 1.04e10T + 1.71e21T^{2} \)
47 \( 1 - 2.14e10T + 5.46e21T^{2} \)
53 \( 1 - 3.32e10T + 2.60e22T^{2} \)
61 \( 1 + 4.78e10T + 1.61e23T^{2} \)
67 \( 1 + 1.27e12T + 5.48e23T^{2} \)
71 \( 1 - 1.28e12T + 1.16e24T^{2} \)
73 \( 1 + 1.47e12T + 1.67e24T^{2} \)
79 \( 1 - 6.13e11T + 4.66e24T^{2} \)
83 \( 1 - 1.96e11T + 8.87e24T^{2} \)
89 \( 1 - 3.68e12T + 2.19e25T^{2} \)
97 \( 1 - 6.38e12T + 6.73e25T^{2} \)
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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

−10.38921586897105111597103920462, −9.215411011877578250326959746389, −8.338696355818733497647095093611, −7.26620772275862320261284587229, −6.18372901404599231978101275914, −4.90208076000056504092940551568, −3.99308446676035837595816989917, −3.25651673704676481421582166054, −2.34827879600838760253496798773, −0.12674835243356723448119483828, 0.12674835243356723448119483828, 2.34827879600838760253496798773, 3.25651673704676481421582166054, 3.99308446676035837595816989917, 4.90208076000056504092940551568, 6.18372901404599231978101275914, 7.26620772275862320261284587229, 8.338696355818733497647095093611, 9.215411011877578250326959746389, 10.38921586897105111597103920462

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