Properties

Label 2-177-1.1-c11-0-107
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  + 81.6·2-s + 243·3-s + 4.61e3·4-s + 4.43e3·5-s + 1.98e4·6-s − 6.76e4·7-s + 2.09e5·8-s + 5.90e4·9-s + 3.61e5·10-s − 5.19e5·11-s + 1.12e6·12-s − 2.46e6·13-s − 5.52e6·14-s + 1.07e6·15-s + 7.64e6·16-s − 4.35e6·17-s + 4.81e6·18-s + 3.11e6·19-s + 2.04e7·20-s − 1.64e7·21-s − 4.24e7·22-s − 1.04e7·23-s + 5.08e7·24-s − 2.92e7·25-s − 2.00e8·26-s + 1.43e7·27-s − 3.12e8·28-s + ⋯
L(s)  = 1  + 1.80·2-s + 0.577·3-s + 2.25·4-s + 0.633·5-s + 1.04·6-s − 1.52·7-s + 2.25·8-s + 0.333·9-s + 1.14·10-s − 0.973·11-s + 1.30·12-s − 1.83·13-s − 2.74·14-s + 0.366·15-s + 1.82·16-s − 0.743·17-s + 0.601·18-s + 0.288·19-s + 1.42·20-s − 0.878·21-s − 1.75·22-s − 0.338·23-s + 1.30·24-s − 0.598·25-s − 3.31·26-s + 0.192·27-s − 3.42·28-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 \( 1 - 81.6T + 2.04e3T^{2} \)
5 \( 1 - 4.43e3T + 4.88e7T^{2} \)
7 \( 1 + 6.76e4T + 1.97e9T^{2} \)
11 \( 1 + 5.19e5T + 2.85e11T^{2} \)
13 \( 1 + 2.46e6T + 1.79e12T^{2} \)
17 \( 1 + 4.35e6T + 3.42e13T^{2} \)
19 \( 1 - 3.11e6T + 1.16e14T^{2} \)
23 \( 1 + 1.04e7T + 9.52e14T^{2} \)
29 \( 1 + 2.13e8T + 1.22e16T^{2} \)
31 \( 1 - 1.99e8T + 2.54e16T^{2} \)
37 \( 1 - 7.90e8T + 1.77e17T^{2} \)
41 \( 1 + 4.32e8T + 5.50e17T^{2} \)
43 \( 1 + 4.93e8T + 9.29e17T^{2} \)
47 \( 1 + 3.47e8T + 2.47e18T^{2} \)
53 \( 1 - 6.14e8T + 9.26e18T^{2} \)
61 \( 1 - 9.09e9T + 4.35e19T^{2} \)
67 \( 1 - 2.20e9T + 1.22e20T^{2} \)
71 \( 1 - 2.45e10T + 2.31e20T^{2} \)
73 \( 1 + 8.48e8T + 3.13e20T^{2} \)
79 \( 1 + 3.82e10T + 7.47e20T^{2} \)
83 \( 1 - 2.31e10T + 1.28e21T^{2} \)
89 \( 1 + 2.16e10T + 2.77e21T^{2} \)
97 \( 1 - 8.63e10T + 7.15e21T^{2} \)
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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

−10.09014138432589865619867390048, −9.565732618650075834269991314595, −7.65970373984780150716526739339, −6.75810095305038444769142678008, −5.82310203516868487709833595964, −4.86375626731128711756181231813, −3.71041225816673383631852555526, −2.63428524620742245389387246781, −2.25217164055018597442110191308, 0, 2.25217164055018597442110191308, 2.63428524620742245389387246781, 3.71041225816673383631852555526, 4.86375626731128711756181231813, 5.82310203516868487709833595964, 6.75810095305038444769142678008, 7.65970373984780150716526739339, 9.565732618650075834269991314595, 10.09014138432589865619867390048

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