Properties

Label 2-177-1.1-c13-0-119
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  + 163.·2-s − 729·3-s + 1.85e4·4-s + 2.04e4·5-s − 1.19e5·6-s − 1.41e5·7-s + 1.68e6·8-s + 5.31e5·9-s + 3.34e6·10-s + 9.53e6·11-s − 1.34e7·12-s − 2.74e7·13-s − 2.31e7·14-s − 1.49e7·15-s + 1.24e8·16-s − 1.51e8·17-s + 8.68e7·18-s − 3.23e8·19-s + 3.79e8·20-s + 1.03e8·21-s + 1.55e9·22-s + 1.57e8·23-s − 1.22e9·24-s − 8.01e8·25-s − 4.48e9·26-s − 3.87e8·27-s − 2.62e9·28-s + ⋯
L(s)  = 1  + 1.80·2-s − 0.577·3-s + 2.26·4-s + 0.586·5-s − 1.04·6-s − 0.455·7-s + 2.27·8-s + 0.333·9-s + 1.05·10-s + 1.62·11-s − 1.30·12-s − 1.57·13-s − 0.821·14-s − 0.338·15-s + 1.84·16-s − 1.51·17-s + 0.601·18-s − 1.57·19-s + 1.32·20-s + 0.262·21-s + 2.92·22-s + 0.222·23-s − 1.31·24-s − 0.656·25-s − 2.84·26-s − 0.192·27-s − 1.02·28-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 \( 1 - 163.T + 8.19e3T^{2} \)
5 \( 1 - 2.04e4T + 1.22e9T^{2} \)
7 \( 1 + 1.41e5T + 9.68e10T^{2} \)
11 \( 1 - 9.53e6T + 3.45e13T^{2} \)
13 \( 1 + 2.74e7T + 3.02e14T^{2} \)
17 \( 1 + 1.51e8T + 9.90e15T^{2} \)
19 \( 1 + 3.23e8T + 4.20e16T^{2} \)
23 \( 1 - 1.57e8T + 5.04e17T^{2} \)
29 \( 1 - 2.89e9T + 1.02e19T^{2} \)
31 \( 1 + 2.87e9T + 2.44e19T^{2} \)
37 \( 1 + 2.22e10T + 2.43e20T^{2} \)
41 \( 1 + 2.17e10T + 9.25e20T^{2} \)
43 \( 1 - 4.89e10T + 1.71e21T^{2} \)
47 \( 1 - 6.68e10T + 5.46e21T^{2} \)
53 \( 1 - 7.25e10T + 2.60e22T^{2} \)
61 \( 1 - 5.91e11T + 1.61e23T^{2} \)
67 \( 1 - 2.90e10T + 5.48e23T^{2} \)
71 \( 1 + 2.23e11T + 1.16e24T^{2} \)
73 \( 1 + 1.85e12T + 1.67e24T^{2} \)
79 \( 1 - 2.22e12T + 4.66e24T^{2} \)
83 \( 1 + 3.73e12T + 8.87e24T^{2} \)
89 \( 1 + 4.14e12T + 2.19e25T^{2} \)
97 \( 1 + 6.55e12T + 6.73e25T^{2} \)
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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

−10.19331962001843908880808034730, −8.996858432132817849461654033637, −6.91756335890837720940526113745, −6.62552321493277046324672577101, −5.65209276116516029153374116355, −4.57642151122946265303221458371, −3.95449853010045314670760380289, −2.51350490754077170278006039872, −1.75864557355029219164378079104, 0, 1.75864557355029219164378079104, 2.51350490754077170278006039872, 3.95449853010045314670760380289, 4.57642151122946265303221458371, 5.65209276116516029153374116355, 6.62552321493277046324672577101, 6.91756335890837720940526113745, 8.996858432132817849461654033637, 10.19331962001843908880808034730

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