Properties

Label 2-177-1.1-c7-0-65
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 10.7·2-s + 27·3-s − 12.1·4-s + 451.·5-s + 290.·6-s − 504.·7-s − 1.50e3·8-s + 729·9-s + 4.86e3·10-s − 8.07e3·11-s − 328.·12-s − 4.67e3·13-s − 5.43e3·14-s + 1.22e4·15-s − 1.46e4·16-s − 1.34e4·17-s + 7.84e3·18-s − 1.13e4·19-s − 5.49e3·20-s − 1.36e4·21-s − 8.68e4·22-s − 6.64e4·23-s − 4.07e4·24-s + 1.26e5·25-s − 5.03e4·26-s + 1.96e4·27-s + 6.13e3·28-s + ⋯
L(s)  = 1  + 0.951·2-s + 0.577·3-s − 0.0949·4-s + 1.61·5-s + 0.549·6-s − 0.556·7-s − 1.04·8-s + 0.333·9-s + 1.53·10-s − 1.82·11-s − 0.0548·12-s − 0.590·13-s − 0.529·14-s + 0.933·15-s − 0.895·16-s − 0.666·17-s + 0.317·18-s − 0.380·19-s − 0.153·20-s − 0.321·21-s − 1.73·22-s − 1.13·23-s − 0.601·24-s + 1.61·25-s − 0.561·26-s + 0.192·27-s + 0.0528·28-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 \( 1 - 10.7T + 128T^{2} \)
5 \( 1 - 451.T + 7.81e4T^{2} \)
7 \( 1 + 504.T + 8.23e5T^{2} \)
11 \( 1 + 8.07e3T + 1.94e7T^{2} \)
13 \( 1 + 4.67e3T + 6.27e7T^{2} \)
17 \( 1 + 1.34e4T + 4.10e8T^{2} \)
19 \( 1 + 1.13e4T + 8.93e8T^{2} \)
23 \( 1 + 6.64e4T + 3.40e9T^{2} \)
29 \( 1 + 1.74e5T + 1.72e10T^{2} \)
31 \( 1 - 6.24e4T + 2.75e10T^{2} \)
37 \( 1 - 2.77e5T + 9.49e10T^{2} \)
41 \( 1 - 5.29e5T + 1.94e11T^{2} \)
43 \( 1 - 6.75e5T + 2.71e11T^{2} \)
47 \( 1 - 1.75e5T + 5.06e11T^{2} \)
53 \( 1 + 3.13e5T + 1.17e12T^{2} \)
61 \( 1 + 3.01e6T + 3.14e12T^{2} \)
67 \( 1 + 8.85e5T + 6.06e12T^{2} \)
71 \( 1 + 1.15e6T + 9.09e12T^{2} \)
73 \( 1 - 3.34e4T + 1.10e13T^{2} \)
79 \( 1 + 6.94e5T + 1.92e13T^{2} \)
83 \( 1 + 6.72e5T + 2.71e13T^{2} \)
89 \( 1 - 7.67e6T + 4.42e13T^{2} \)
97 \( 1 - 4.89e6T + 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.71943827056050149744978051259, −9.762663049357801183911139672402, −9.150154511026007440319140282806, −7.75953242891771597570719396529, −6.22215178088750118984607578552, −5.52560361283704725825879079218, −4.41860329384364271837411016948, −2.83207764001527264668292232126, −2.20751479641618996134483535616, 0, 2.20751479641618996134483535616, 2.83207764001527264668292232126, 4.41860329384364271837411016948, 5.52560361283704725825879079218, 6.22215178088750118984607578552, 7.75953242891771597570719396529, 9.150154511026007440319140282806, 9.762663049357801183911139672402, 10.71943827056050149744978051259

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