Properties

Label 2-177-1.1-c9-0-52
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  − 36.0·2-s − 81·3-s + 787.·4-s − 242.·5-s + 2.91e3·6-s + 7.61e3·7-s − 9.92e3·8-s + 6.56e3·9-s + 8.72e3·10-s + 1.70e4·11-s − 6.37e4·12-s + 6.82e4·13-s − 2.74e5·14-s + 1.96e4·15-s − 4.54e4·16-s + 3.21e5·17-s − 2.36e5·18-s − 8.03e5·19-s − 1.90e5·20-s − 6.16e5·21-s − 6.16e5·22-s + 9.95e5·23-s + 8.03e5·24-s − 1.89e6·25-s − 2.46e6·26-s − 5.31e5·27-s + 5.99e6·28-s + ⋯
L(s)  = 1  − 1.59·2-s − 0.577·3-s + 1.53·4-s − 0.173·5-s + 0.919·6-s + 1.19·7-s − 0.856·8-s + 0.333·9-s + 0.276·10-s + 0.352·11-s − 0.887·12-s + 0.662·13-s − 1.90·14-s + 0.100·15-s − 0.173·16-s + 0.934·17-s − 0.530·18-s − 1.41·19-s − 0.266·20-s − 0.691·21-s − 0.560·22-s + 0.741·23-s + 0.494·24-s − 0.969·25-s − 1.05·26-s − 0.192·27-s + 1.84·28-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 \( 1 + 36.0T + 512T^{2} \)
5 \( 1 + 242.T + 1.95e6T^{2} \)
7 \( 1 - 7.61e3T + 4.03e7T^{2} \)
11 \( 1 - 1.70e4T + 2.35e9T^{2} \)
13 \( 1 - 6.82e4T + 1.06e10T^{2} \)
17 \( 1 - 3.21e5T + 1.18e11T^{2} \)
19 \( 1 + 8.03e5T + 3.22e11T^{2} \)
23 \( 1 - 9.95e5T + 1.80e12T^{2} \)
29 \( 1 + 4.98e6T + 1.45e13T^{2} \)
31 \( 1 + 3.39e6T + 2.64e13T^{2} \)
37 \( 1 - 2.66e6T + 1.29e14T^{2} \)
41 \( 1 - 2.95e6T + 3.27e14T^{2} \)
43 \( 1 + 4.26e7T + 5.02e14T^{2} \)
47 \( 1 - 2.32e7T + 1.11e15T^{2} \)
53 \( 1 - 6.00e7T + 3.29e15T^{2} \)
61 \( 1 + 7.21e6T + 1.16e16T^{2} \)
67 \( 1 - 2.28e8T + 2.72e16T^{2} \)
71 \( 1 + 5.89e7T + 4.58e16T^{2} \)
73 \( 1 - 1.06e8T + 5.88e16T^{2} \)
79 \( 1 + 3.06e7T + 1.19e17T^{2} \)
83 \( 1 - 2.84e8T + 1.86e17T^{2} \)
89 \( 1 - 7.19e7T + 3.50e17T^{2} \)
97 \( 1 + 8.66e8T + 7.60e17T^{2} \)
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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

−10.57996731547663411393322093311, −9.453284089549150832779642255136, −8.453118974027657075125485278431, −7.74877555624938126832521756088, −6.69392131495967461304153289209, −5.40539402975371061501287961873, −3.99521721486214325205103377160, −1.96222209286129687317195613231, −1.17879065267621602075330909566, 0, 1.17879065267621602075330909566, 1.96222209286129687317195613231, 3.99521721486214325205103377160, 5.40539402975371061501287961873, 6.69392131495967461304153289209, 7.74877555624938126832521756088, 8.453118974027657075125485278431, 9.453284089549150832779642255136, 10.57996731547663411393322093311

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