Properties

Label 2-177-1.1-c11-0-49
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  − 18.7·2-s + 243·3-s − 1.69e3·4-s − 6.32e3·5-s − 4.55e3·6-s − 5.96e4·7-s + 7.02e4·8-s + 5.90e4·9-s + 1.18e5·10-s − 6.23e5·11-s − 4.12e5·12-s + 5.28e5·13-s + 1.11e6·14-s − 1.53e6·15-s + 2.15e6·16-s + 8.97e6·17-s − 1.10e6·18-s − 1.60e7·19-s + 1.07e7·20-s − 1.44e7·21-s + 1.16e7·22-s + 2.78e7·23-s + 1.70e7·24-s − 8.80e6·25-s − 9.91e6·26-s + 1.43e7·27-s + 1.01e8·28-s + ⋯
L(s)  = 1  − 0.414·2-s + 0.577·3-s − 0.828·4-s − 0.905·5-s − 0.239·6-s − 1.34·7-s + 0.757·8-s + 0.333·9-s + 0.375·10-s − 1.16·11-s − 0.478·12-s + 0.394·13-s + 0.556·14-s − 0.522·15-s + 0.514·16-s + 1.53·17-s − 0.138·18-s − 1.48·19-s + 0.749·20-s − 0.774·21-s + 0.483·22-s + 0.901·23-s + 0.437·24-s − 0.180·25-s − 0.163·26-s + 0.192·27-s + 1.11·28-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 \( 1 + 18.7T + 2.04e3T^{2} \)
5 \( 1 + 6.32e3T + 4.88e7T^{2} \)
7 \( 1 + 5.96e4T + 1.97e9T^{2} \)
11 \( 1 + 6.23e5T + 2.85e11T^{2} \)
13 \( 1 - 5.28e5T + 1.79e12T^{2} \)
17 \( 1 - 8.97e6T + 3.42e13T^{2} \)
19 \( 1 + 1.60e7T + 1.16e14T^{2} \)
23 \( 1 - 2.78e7T + 9.52e14T^{2} \)
29 \( 1 - 1.64e8T + 1.22e16T^{2} \)
31 \( 1 + 1.62e7T + 2.54e16T^{2} \)
37 \( 1 - 2.65e8T + 1.77e17T^{2} \)
41 \( 1 - 8.26e8T + 5.50e17T^{2} \)
43 \( 1 - 1.62e9T + 9.29e17T^{2} \)
47 \( 1 + 7.37e8T + 2.47e18T^{2} \)
53 \( 1 + 5.01e9T + 9.26e18T^{2} \)
61 \( 1 - 1.38e9T + 4.35e19T^{2} \)
67 \( 1 + 1.90e10T + 1.22e20T^{2} \)
71 \( 1 - 1.84e10T + 2.31e20T^{2} \)
73 \( 1 + 7.34e9T + 3.13e20T^{2} \)
79 \( 1 + 8.78e9T + 7.47e20T^{2} \)
83 \( 1 - 4.09e10T + 1.28e21T^{2} \)
89 \( 1 - 3.82e10T + 2.77e21T^{2} \)
97 \( 1 + 9.18e10T + 7.15e21T^{2} \)
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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

−10.02352904367241529242997847769, −9.125983790271290278230427017091, −8.150116463363102445845965128596, −7.53863535892640204058937630944, −6.09527797947349701965201425308, −4.64708858529583714115838749999, −3.63688057606525250219467495061, −2.77326286459154685220129501346, −0.903524047917948857065650523730, 0, 0.903524047917948857065650523730, 2.77326286459154685220129501346, 3.63688057606525250219467495061, 4.64708858529583714115838749999, 6.09527797947349701965201425308, 7.53863535892640204058937630944, 8.150116463363102445845965128596, 9.125983790271290278230427017091, 10.02352904367241529242997847769

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