Properties

Label 2-177-1.1-c9-0-66
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  + 16.0·2-s − 81·3-s − 253.·4-s + 2.04e3·5-s − 1.30e3·6-s − 8.88e3·7-s − 1.23e4·8-s + 6.56e3·9-s + 3.29e4·10-s + 7.09e4·11-s + 2.05e4·12-s − 5.77e4·13-s − 1.42e5·14-s − 1.65e5·15-s − 6.82e4·16-s + 5.74e5·17-s + 1.05e5·18-s − 6.75e5·19-s − 5.19e5·20-s + 7.19e5·21-s + 1.14e6·22-s + 3.25e5·23-s + 9.96e5·24-s + 2.24e6·25-s − 9.28e5·26-s − 5.31e5·27-s + 2.25e6·28-s + ⋯
L(s)  = 1  + 0.710·2-s − 0.577·3-s − 0.494·4-s + 1.46·5-s − 0.410·6-s − 1.39·7-s − 1.06·8-s + 0.333·9-s + 1.04·10-s + 1.46·11-s + 0.285·12-s − 0.560·13-s − 0.993·14-s − 0.846·15-s − 0.260·16-s + 1.66·17-s + 0.236·18-s − 1.18·19-s − 0.725·20-s + 0.807·21-s + 1.03·22-s + 0.242·23-s + 0.613·24-s + 1.14·25-s − 0.398·26-s − 0.192·27-s + 0.691·28-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 \( 1 - 16.0T + 512T^{2} \)
5 \( 1 - 2.04e3T + 1.95e6T^{2} \)
7 \( 1 + 8.88e3T + 4.03e7T^{2} \)
11 \( 1 - 7.09e4T + 2.35e9T^{2} \)
13 \( 1 + 5.77e4T + 1.06e10T^{2} \)
17 \( 1 - 5.74e5T + 1.18e11T^{2} \)
19 \( 1 + 6.75e5T + 3.22e11T^{2} \)
23 \( 1 - 3.25e5T + 1.80e12T^{2} \)
29 \( 1 + 2.59e6T + 1.45e13T^{2} \)
31 \( 1 - 3.69e6T + 2.64e13T^{2} \)
37 \( 1 + 1.59e7T + 1.29e14T^{2} \)
41 \( 1 - 2.01e7T + 3.27e14T^{2} \)
43 \( 1 + 2.99e7T + 5.02e14T^{2} \)
47 \( 1 - 1.30e7T + 1.11e15T^{2} \)
53 \( 1 + 4.33e7T + 3.29e15T^{2} \)
61 \( 1 - 1.48e8T + 1.16e16T^{2} \)
67 \( 1 - 2.62e7T + 2.72e16T^{2} \)
71 \( 1 + 3.59e8T + 4.58e16T^{2} \)
73 \( 1 + 3.34e8T + 5.88e16T^{2} \)
79 \( 1 + 6.43e8T + 1.19e17T^{2} \)
83 \( 1 + 3.84e8T + 1.86e17T^{2} \)
89 \( 1 - 3.95e8T + 3.50e17T^{2} \)
97 \( 1 + 3.41e8T + 7.60e17T^{2} \)
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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

−10.17262957111010208663882235369, −9.702272200621924359620198631095, −8.892895781164189896186174955952, −6.80239637399639461954738889151, −6.09399456715979522479192897266, −5.41442539533712147589523288924, −4.06454211634504502563428164360, −2.96015470622633966760037488215, −1.37993359438436474199849260643, 0, 1.37993359438436474199849260643, 2.96015470622633966760037488215, 4.06454211634504502563428164360, 5.41442539533712147589523288924, 6.09399456715979522479192897266, 6.80239637399639461954738889151, 8.892895781164189896186174955952, 9.702272200621924359620198631095, 10.17262957111010208663882235369

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