Properties

Label 2-177-1.1-c9-0-34
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  + 7.87·2-s − 81·3-s − 450.·4-s − 1.39e3·5-s − 637.·6-s − 8.51e3·7-s − 7.57e3·8-s + 6.56e3·9-s − 1.09e4·10-s − 3.27e4·11-s + 3.64e4·12-s + 1.73e5·13-s − 6.70e4·14-s + 1.12e5·15-s + 1.70e5·16-s + 5.57e5·17-s + 5.16e4·18-s + 3.29e5·19-s + 6.27e5·20-s + 6.90e5·21-s − 2.58e5·22-s + 1.21e5·23-s + 6.13e5·24-s − 7.93e3·25-s + 1.36e6·26-s − 5.31e5·27-s + 3.83e6·28-s + ⋯
L(s)  = 1  + 0.347·2-s − 0.577·3-s − 0.878·4-s − 0.997·5-s − 0.200·6-s − 1.34·7-s − 0.653·8-s + 0.333·9-s − 0.347·10-s − 0.675·11-s + 0.507·12-s + 1.68·13-s − 0.466·14-s + 0.576·15-s + 0.651·16-s + 1.61·17-s + 0.115·18-s + 0.579·19-s + 0.877·20-s + 0.774·21-s − 0.234·22-s + 0.0904·23-s + 0.377·24-s − 0.00406·25-s + 0.585·26-s − 0.192·27-s + 1.17·28-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 \( 1 - 7.87T + 512T^{2} \)
5 \( 1 + 1.39e3T + 1.95e6T^{2} \)
7 \( 1 + 8.51e3T + 4.03e7T^{2} \)
11 \( 1 + 3.27e4T + 2.35e9T^{2} \)
13 \( 1 - 1.73e5T + 1.06e10T^{2} \)
17 \( 1 - 5.57e5T + 1.18e11T^{2} \)
19 \( 1 - 3.29e5T + 3.22e11T^{2} \)
23 \( 1 - 1.21e5T + 1.80e12T^{2} \)
29 \( 1 + 5.99e3T + 1.45e13T^{2} \)
31 \( 1 + 7.64e6T + 2.64e13T^{2} \)
37 \( 1 + 7.81e6T + 1.29e14T^{2} \)
41 \( 1 + 1.42e7T + 3.27e14T^{2} \)
43 \( 1 - 1.08e7T + 5.02e14T^{2} \)
47 \( 1 - 1.53e7T + 1.11e15T^{2} \)
53 \( 1 + 2.75e6T + 3.29e15T^{2} \)
61 \( 1 - 1.86e8T + 1.16e16T^{2} \)
67 \( 1 + 3.01e7T + 2.72e16T^{2} \)
71 \( 1 + 9.51e6T + 4.58e16T^{2} \)
73 \( 1 - 2.10e8T + 5.88e16T^{2} \)
79 \( 1 - 3.67e7T + 1.19e17T^{2} \)
83 \( 1 - 8.97e7T + 1.86e17T^{2} \)
89 \( 1 + 8.83e7T + 3.50e17T^{2} \)
97 \( 1 + 1.72e8T + 7.60e17T^{2} \)
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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

−10.51943504292445399391038071833, −9.588473579103524079470338972229, −8.478580185122227196015618801572, −7.40953709281823472531161443146, −6.05178511958865884993481501729, −5.27581140267131733761806420312, −3.70407688053437459868698706898, −3.43263675539239849846614340091, −0.914320369362796677828234804212, 0, 0.914320369362796677828234804212, 3.43263675539239849846614340091, 3.70407688053437459868698706898, 5.27581140267131733761806420312, 6.05178511958865884993481501729, 7.40953709281823472531161443146, 8.478580185122227196015618801572, 9.588473579103524079470338972229, 10.51943504292445399391038071833

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