Properties

Label 2-177-1.1-c9-0-77
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  + 27.0·2-s + 81·3-s + 222.·4-s − 1.51e3·5-s + 2.19e3·6-s + 2.21e3·7-s − 7.84e3·8-s + 6.56e3·9-s − 4.11e4·10-s + 2.04e4·11-s + 1.80e4·12-s + 1.46e5·13-s + 5.98e4·14-s − 1.22e5·15-s − 3.26e5·16-s + 1.06e4·17-s + 1.77e5·18-s − 3.63e5·19-s − 3.37e5·20-s + 1.79e5·21-s + 5.55e5·22-s + 1.47e6·23-s − 6.35e5·24-s + 3.51e5·25-s + 3.97e6·26-s + 5.31e5·27-s + 4.91e5·28-s + ⋯
L(s)  = 1  + 1.19·2-s + 0.577·3-s + 0.434·4-s − 1.08·5-s + 0.691·6-s + 0.347·7-s − 0.677·8-s + 0.333·9-s − 1.30·10-s + 0.421·11-s + 0.250·12-s + 1.42·13-s + 0.416·14-s − 0.627·15-s − 1.24·16-s + 0.0309·17-s + 0.399·18-s − 0.639·19-s − 0.471·20-s + 0.200·21-s + 0.505·22-s + 1.10·23-s − 0.391·24-s + 0.179·25-s + 1.70·26-s + 0.192·27-s + 0.151·28-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 \( 1 - 27.0T + 512T^{2} \)
5 \( 1 + 1.51e3T + 1.95e6T^{2} \)
7 \( 1 - 2.21e3T + 4.03e7T^{2} \)
11 \( 1 - 2.04e4T + 2.35e9T^{2} \)
13 \( 1 - 1.46e5T + 1.06e10T^{2} \)
17 \( 1 - 1.06e4T + 1.18e11T^{2} \)
19 \( 1 + 3.63e5T + 3.22e11T^{2} \)
23 \( 1 - 1.47e6T + 1.80e12T^{2} \)
29 \( 1 + 7.12e6T + 1.45e13T^{2} \)
31 \( 1 + 8.92e6T + 2.64e13T^{2} \)
37 \( 1 + 3.34e6T + 1.29e14T^{2} \)
41 \( 1 - 1.53e7T + 3.27e14T^{2} \)
43 \( 1 + 2.67e7T + 5.02e14T^{2} \)
47 \( 1 + 5.05e7T + 1.11e15T^{2} \)
53 \( 1 + 5.26e7T + 3.29e15T^{2} \)
61 \( 1 - 3.72e7T + 1.16e16T^{2} \)
67 \( 1 + 1.12e8T + 2.72e16T^{2} \)
71 \( 1 + 2.42e8T + 4.58e16T^{2} \)
73 \( 1 + 1.68e7T + 5.88e16T^{2} \)
79 \( 1 - 5.59e8T + 1.19e17T^{2} \)
83 \( 1 + 7.22e8T + 1.86e17T^{2} \)
89 \( 1 - 5.52e8T + 3.50e17T^{2} \)
97 \( 1 - 3.40e8T + 7.60e17T^{2} \)
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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

−11.05978530931261986640002503569, −9.235307592222706752536316548036, −8.475298770335915483788475701648, −7.35571859904154577793336377002, −6.15061961494707400355735412344, −4.88088408837944115247741948899, −3.79367108168903866135994865499, −3.39504897339055304746064896815, −1.68357125683118548993944331613, 0, 1.68357125683118548993944331613, 3.39504897339055304746064896815, 3.79367108168903866135994865499, 4.88088408837944115247741948899, 6.15061961494707400355735412344, 7.35571859904154577793336377002, 8.475298770335915483788475701648, 9.235307592222706752536316548036, 11.05978530931261986640002503569

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