Properties

Label 2-177-1.1-c9-0-61
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  − 35.8·2-s + 81·3-s + 773.·4-s + 1.88e3·5-s − 2.90e3·6-s − 9.73e3·7-s − 9.35e3·8-s + 6.56e3·9-s − 6.74e4·10-s − 6.16e4·11-s + 6.26e4·12-s + 1.69e5·13-s + 3.49e5·14-s + 1.52e5·15-s − 6.03e4·16-s + 5.60e5·17-s − 2.35e5·18-s − 6.59e5·19-s + 1.45e6·20-s − 7.88e5·21-s + 2.20e6·22-s − 5.35e5·23-s − 7.58e5·24-s + 1.58e6·25-s − 6.09e6·26-s + 5.31e5·27-s − 7.52e6·28-s + ⋯
L(s)  = 1  − 1.58·2-s + 0.577·3-s + 1.50·4-s + 1.34·5-s − 0.914·6-s − 1.53·7-s − 0.807·8-s + 0.333·9-s − 2.13·10-s − 1.26·11-s + 0.871·12-s + 1.65·13-s + 2.42·14-s + 0.777·15-s − 0.230·16-s + 1.62·17-s − 0.528·18-s − 1.16·19-s + 2.03·20-s − 0.885·21-s + 2.01·22-s − 0.399·23-s − 0.466·24-s + 0.811·25-s − 2.61·26-s + 0.192·27-s − 2.31·28-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 \( 1 + 35.8T + 512T^{2} \)
5 \( 1 - 1.88e3T + 1.95e6T^{2} \)
7 \( 1 + 9.73e3T + 4.03e7T^{2} \)
11 \( 1 + 6.16e4T + 2.35e9T^{2} \)
13 \( 1 - 1.69e5T + 1.06e10T^{2} \)
17 \( 1 - 5.60e5T + 1.18e11T^{2} \)
19 \( 1 + 6.59e5T + 3.22e11T^{2} \)
23 \( 1 + 5.35e5T + 1.80e12T^{2} \)
29 \( 1 - 1.76e6T + 1.45e13T^{2} \)
31 \( 1 + 7.35e6T + 2.64e13T^{2} \)
37 \( 1 + 1.11e7T + 1.29e14T^{2} \)
41 \( 1 - 9.17e6T + 3.27e14T^{2} \)
43 \( 1 - 1.27e7T + 5.02e14T^{2} \)
47 \( 1 - 7.13e6T + 1.11e15T^{2} \)
53 \( 1 + 1.07e8T + 3.29e15T^{2} \)
61 \( 1 + 2.18e7T + 1.16e16T^{2} \)
67 \( 1 - 3.04e8T + 2.72e16T^{2} \)
71 \( 1 + 2.46e8T + 4.58e16T^{2} \)
73 \( 1 + 9.89e7T + 5.88e16T^{2} \)
79 \( 1 + 1.89e7T + 1.19e17T^{2} \)
83 \( 1 + 3.82e8T + 1.86e17T^{2} \)
89 \( 1 - 8.54e8T + 3.50e17T^{2} \)
97 \( 1 - 1.05e9T + 7.60e17T^{2} \)
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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

−10.22548122816380581976637401980, −9.503338481368873628008749673642, −8.745969514667716660657796068595, −7.76353927154633557407048731893, −6.51661634446445038042991563437, −5.73814399657382586221977459580, −3.42291068132849781568495360730, −2.34025153946217258887890814080, −1.29842084976226338949339927403, 0, 1.29842084976226338949339927403, 2.34025153946217258887890814080, 3.42291068132849781568495360730, 5.73814399657382586221977459580, 6.51661634446445038042991563437, 7.76353927154633557407048731893, 8.745969514667716660657796068595, 9.503338481368873628008749673642, 10.22548122816380581976637401980

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