Properties

Label 2-177-1.1-c9-0-68
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  + 24.6·2-s − 81·3-s + 96.9·4-s + 1.35e3·5-s − 1.99e3·6-s − 4.83e3·7-s − 1.02e4·8-s + 6.56e3·9-s + 3.33e4·10-s + 1.73e4·11-s − 7.85e3·12-s + 1.56e5·13-s − 1.19e5·14-s − 1.09e5·15-s − 3.02e5·16-s − 3.58e5·17-s + 1.61e5·18-s + 7.77e4·19-s + 1.31e5·20-s + 3.91e5·21-s + 4.29e5·22-s + 1.04e6·23-s + 8.29e5·24-s − 1.25e5·25-s + 3.85e6·26-s − 5.31e5·27-s − 4.68e5·28-s + ⋯
L(s)  = 1  + 1.09·2-s − 0.577·3-s + 0.189·4-s + 0.967·5-s − 0.629·6-s − 0.760·7-s − 0.884·8-s + 0.333·9-s + 1.05·10-s + 0.358·11-s − 0.109·12-s + 1.51·13-s − 0.829·14-s − 0.558·15-s − 1.15·16-s − 1.04·17-s + 0.363·18-s + 0.136·19-s + 0.183·20-s + 0.439·21-s + 0.390·22-s + 0.775·23-s + 0.510·24-s − 0.0644·25-s + 1.65·26-s − 0.192·27-s − 0.143·28-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 \( 1 - 24.6T + 512T^{2} \)
5 \( 1 - 1.35e3T + 1.95e6T^{2} \)
7 \( 1 + 4.83e3T + 4.03e7T^{2} \)
11 \( 1 - 1.73e4T + 2.35e9T^{2} \)
13 \( 1 - 1.56e5T + 1.06e10T^{2} \)
17 \( 1 + 3.58e5T + 1.18e11T^{2} \)
19 \( 1 - 7.77e4T + 3.22e11T^{2} \)
23 \( 1 - 1.04e6T + 1.80e12T^{2} \)
29 \( 1 - 4.51e6T + 1.45e13T^{2} \)
31 \( 1 + 9.81e6T + 2.64e13T^{2} \)
37 \( 1 - 5.36e6T + 1.29e14T^{2} \)
41 \( 1 + 1.59e7T + 3.27e14T^{2} \)
43 \( 1 + 1.04e7T + 5.02e14T^{2} \)
47 \( 1 + 2.40e7T + 1.11e15T^{2} \)
53 \( 1 - 2.96e7T + 3.29e15T^{2} \)
61 \( 1 + 1.50e8T + 1.16e16T^{2} \)
67 \( 1 + 1.05e8T + 2.72e16T^{2} \)
71 \( 1 + 4.08e8T + 4.58e16T^{2} \)
73 \( 1 - 3.83e8T + 5.88e16T^{2} \)
79 \( 1 - 2.66e8T + 1.19e17T^{2} \)
83 \( 1 + 7.10e8T + 1.86e17T^{2} \)
89 \( 1 + 9.56e8T + 3.50e17T^{2} \)
97 \( 1 - 8.46e8T + 7.60e17T^{2} \)
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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

−10.72390985534786195440482374018, −9.514157089072407277559305759201, −8.756989721892459505815895470990, −6.69561359933512112306909021490, −6.16645587966291002421469605125, −5.28549596830170248841850180847, −4.09824536444775189461111915728, −3.04352096199006403054110174327, −1.52103987204814498548091550181, 0, 1.52103987204814498548091550181, 3.04352096199006403054110174327, 4.09824536444775189461111915728, 5.28549596830170248841850180847, 6.16645587966291002421469605125, 6.69561359933512112306909021490, 8.756989721892459505815895470990, 9.514157089072407277559305759201, 10.72390985534786195440482374018

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