Properties

Label 2-177-1.1-c11-0-92
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  − 43.6·2-s − 243·3-s − 142.·4-s + 7.12e3·5-s + 1.06e4·6-s + 5.03e4·7-s + 9.56e4·8-s + 5.90e4·9-s − 3.11e5·10-s + 3.65e5·11-s + 3.45e4·12-s + 6.75e5·13-s − 2.19e6·14-s − 1.73e6·15-s − 3.88e6·16-s + 5.75e6·17-s − 2.57e6·18-s + 6.02e6·19-s − 1.01e6·20-s − 1.22e7·21-s − 1.59e7·22-s − 3.76e7·23-s − 2.32e7·24-s + 1.92e6·25-s − 2.94e7·26-s − 1.43e7·27-s − 7.14e6·28-s + ⋯
L(s)  = 1  − 0.964·2-s − 0.577·3-s − 0.0693·4-s + 1.01·5-s + 0.556·6-s + 1.13·7-s + 1.03·8-s + 0.333·9-s − 0.983·10-s + 0.683·11-s + 0.0400·12-s + 0.504·13-s − 1.09·14-s − 0.588·15-s − 0.925·16-s + 0.983·17-s − 0.321·18-s + 0.558·19-s − 0.0707·20-s − 0.653·21-s − 0.659·22-s − 1.22·23-s − 0.595·24-s + 0.0395·25-s − 0.486·26-s − 0.192·27-s − 0.0785·28-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 \( 1 + 43.6T + 2.04e3T^{2} \)
5 \( 1 - 7.12e3T + 4.88e7T^{2} \)
7 \( 1 - 5.03e4T + 1.97e9T^{2} \)
11 \( 1 - 3.65e5T + 2.85e11T^{2} \)
13 \( 1 - 6.75e5T + 1.79e12T^{2} \)
17 \( 1 - 5.75e6T + 3.42e13T^{2} \)
19 \( 1 - 6.02e6T + 1.16e14T^{2} \)
23 \( 1 + 3.76e7T + 9.52e14T^{2} \)
29 \( 1 + 1.23e8T + 1.22e16T^{2} \)
31 \( 1 + 3.05e8T + 2.54e16T^{2} \)
37 \( 1 - 4.58e8T + 1.77e17T^{2} \)
41 \( 1 + 1.12e9T + 5.50e17T^{2} \)
43 \( 1 + 1.10e9T + 9.29e17T^{2} \)
47 \( 1 - 1.34e9T + 2.47e18T^{2} \)
53 \( 1 + 5.46e9T + 9.26e18T^{2} \)
61 \( 1 - 2.87e8T + 4.35e19T^{2} \)
67 \( 1 - 6.29e9T + 1.22e20T^{2} \)
71 \( 1 + 1.39e10T + 2.31e20T^{2} \)
73 \( 1 + 2.18e10T + 3.13e20T^{2} \)
79 \( 1 + 1.00e9T + 7.47e20T^{2} \)
83 \( 1 + 8.58e9T + 1.28e21T^{2} \)
89 \( 1 + 9.54e10T + 2.77e21T^{2} \)
97 \( 1 - 7.20e10T + 7.15e21T^{2} \)
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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

−9.978671725754553581959061932006, −9.346915262629965641106380531837, −8.239539171201266322383112587944, −7.34619480532828563760643397217, −5.93386351105562095756413467677, −5.13538167729612863734370754475, −3.86994130079126221844779874438, −1.68512780134285768721516329667, −1.43832344459819988523119572844, 0, 1.43832344459819988523119572844, 1.68512780134285768721516329667, 3.86994130079126221844779874438, 5.13538167729612863734370754475, 5.93386351105562095756413467677, 7.34619480532828563760643397217, 8.239539171201266322383112587944, 9.346915262629965641106380531837, 9.978671725754553581959061932006

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