Properties

Label 2-177-1.1-c11-0-106
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 75.6·2-s + 243·3-s + 3.67e3·4-s − 5.74e3·5-s + 1.83e4·6-s + 1.09e4·7-s + 1.22e5·8-s + 5.90e4·9-s − 4.34e5·10-s − 3.43e5·11-s + 8.92e5·12-s + 1.01e6·13-s + 8.27e5·14-s − 1.39e6·15-s + 1.77e6·16-s − 6.50e6·17-s + 4.46e6·18-s − 2.12e7·19-s − 2.10e7·20-s + 2.65e6·21-s − 2.59e7·22-s + 8.47e6·23-s + 2.98e7·24-s − 1.58e7·25-s + 7.67e7·26-s + 1.43e7·27-s + 4.01e7·28-s + ⋯
L(s)  = 1  + 1.67·2-s + 0.577·3-s + 1.79·4-s − 0.821·5-s + 0.965·6-s + 0.246·7-s + 1.32·8-s + 0.333·9-s − 1.37·10-s − 0.642·11-s + 1.03·12-s + 0.757·13-s + 0.411·14-s − 0.474·15-s + 0.424·16-s − 1.11·17-s + 0.557·18-s − 1.96·19-s − 1.47·20-s + 0.142·21-s − 1.07·22-s + 0.274·23-s + 0.766·24-s − 0.325·25-s + 1.26·26-s + 0.192·27-s + 0.441·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 243T \)
59 \( 1 + 7.14e8T \)
good2 \( 1 - 75.6T + 2.04e3T^{2} \)
5 \( 1 + 5.74e3T + 4.88e7T^{2} \)
7 \( 1 - 1.09e4T + 1.97e9T^{2} \)
11 \( 1 + 3.43e5T + 2.85e11T^{2} \)
13 \( 1 - 1.01e6T + 1.79e12T^{2} \)
17 \( 1 + 6.50e6T + 3.42e13T^{2} \)
19 \( 1 + 2.12e7T + 1.16e14T^{2} \)
23 \( 1 - 8.47e6T + 9.52e14T^{2} \)
29 \( 1 + 7.43e7T + 1.22e16T^{2} \)
31 \( 1 - 6.43e7T + 2.54e16T^{2} \)
37 \( 1 + 2.96e8T + 1.77e17T^{2} \)
41 \( 1 - 1.07e9T + 5.50e17T^{2} \)
43 \( 1 - 5.20e8T + 9.29e17T^{2} \)
47 \( 1 - 1.66e9T + 2.47e18T^{2} \)
53 \( 1 + 1.12e9T + 9.26e18T^{2} \)
61 \( 1 + 1.12e10T + 4.35e19T^{2} \)
67 \( 1 - 5.98e9T + 1.22e20T^{2} \)
71 \( 1 - 2.52e10T + 2.31e20T^{2} \)
73 \( 1 + 1.92e10T + 3.13e20T^{2} \)
79 \( 1 + 3.62e10T + 7.47e20T^{2} \)
83 \( 1 + 6.51e10T + 1.28e21T^{2} \)
89 \( 1 + 3.09e10T + 2.77e21T^{2} \)
97 \( 1 - 9.89e10T + 7.15e21T^{2} \)
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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

−10.74980852602913225031284943180, −8.922424737105785610237429974854, −7.960938034557753569112773092051, −6.85132240581884021453426782646, −5.84052221358806383579697131539, −4.45570207367355980701594895689, −4.03343371064337642146130386425, −2.85355193395909264017083350580, −1.91172462930712744546163525501, 0, 1.91172462930712744546163525501, 2.85355193395909264017083350580, 4.03343371064337642146130386425, 4.45570207367355980701594895689, 5.84052221358806383579697131539, 6.85132240581884021453426782646, 7.960938034557753569112773092051, 8.922424737105785610237429974854, 10.74980852602913225031284943180

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