Properties

Label 2-177-1.1-c9-0-35
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.43·2-s − 81·3-s − 492.·4-s + 339.·5-s + 359.·6-s − 1.23e4·7-s + 4.45e3·8-s + 6.56e3·9-s − 1.50e3·10-s + 5.29e4·11-s + 3.98e4·12-s − 1.28e5·13-s + 5.47e4·14-s − 2.74e4·15-s + 2.32e5·16-s − 5.63e5·17-s − 2.91e4·18-s + 8.33e5·19-s − 1.67e5·20-s + 9.99e5·21-s − 2.35e5·22-s + 2.12e6·23-s − 3.60e5·24-s − 1.83e6·25-s + 5.68e5·26-s − 5.31e5·27-s + 6.07e6·28-s + ⋯
L(s)  = 1  − 0.196·2-s − 0.577·3-s − 0.961·4-s + 0.242·5-s + 0.113·6-s − 1.94·7-s + 0.384·8-s + 0.333·9-s − 0.0476·10-s + 1.09·11-s + 0.555·12-s − 1.24·13-s + 0.380·14-s − 0.140·15-s + 0.886·16-s − 1.63·17-s − 0.0653·18-s + 1.46·19-s − 0.233·20-s + 1.12·21-s − 0.213·22-s + 1.58·23-s − 0.222·24-s − 0.940·25-s + 0.243·26-s − 0.192·27-s + 1.86·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 + 4.43T + 512T^{2} \)
5 \( 1 - 339.T + 1.95e6T^{2} \)
7 \( 1 + 1.23e4T + 4.03e7T^{2} \)
11 \( 1 - 5.29e4T + 2.35e9T^{2} \)
13 \( 1 + 1.28e5T + 1.06e10T^{2} \)
17 \( 1 + 5.63e5T + 1.18e11T^{2} \)
19 \( 1 - 8.33e5T + 3.22e11T^{2} \)
23 \( 1 - 2.12e6T + 1.80e12T^{2} \)
29 \( 1 - 5.73e6T + 1.45e13T^{2} \)
31 \( 1 - 1.18e6T + 2.64e13T^{2} \)
37 \( 1 - 5.87e6T + 1.29e14T^{2} \)
41 \( 1 + 2.15e7T + 3.27e14T^{2} \)
43 \( 1 - 2.71e7T + 5.02e14T^{2} \)
47 \( 1 + 2.75e7T + 1.11e15T^{2} \)
53 \( 1 - 2.71e7T + 3.29e15T^{2} \)
61 \( 1 - 9.59e7T + 1.16e16T^{2} \)
67 \( 1 - 1.74e8T + 2.72e16T^{2} \)
71 \( 1 - 1.39e8T + 4.58e16T^{2} \)
73 \( 1 + 1.56e8T + 5.88e16T^{2} \)
79 \( 1 + 5.55e8T + 1.19e17T^{2} \)
83 \( 1 + 2.03e8T + 1.86e17T^{2} \)
89 \( 1 - 1.86e7T + 3.50e17T^{2} \)
97 \( 1 + 6.59e8T + 7.60e17T^{2} \)
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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

−10.10667691004869562219689641997, −9.586907351882043050613468931170, −8.911958315144938810064479138300, −7.13198363928228433376742894583, −6.43438942343272189314830148242, −5.18154722202721145932180333421, −4.07310895332002455819252663385, −2.84712860670202482851661678303, −0.921901300041829951787095264300, 0, 0.921901300041829951787095264300, 2.84712860670202482851661678303, 4.07310895332002455819252663385, 5.18154722202721145932180333421, 6.43438942343272189314830148242, 7.13198363928228433376742894583, 8.911958315144938810064479138300, 9.586907351882043050613468931170, 10.10667691004869562219689641997

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