Properties

Label 2-177-1.1-c11-0-57
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  − 87.0·2-s − 243·3-s + 5.52e3·4-s − 497.·5-s + 2.11e4·6-s + 1.08e4·7-s − 3.02e5·8-s + 5.90e4·9-s + 4.32e4·10-s + 3.10e5·11-s − 1.34e6·12-s − 1.09e6·13-s − 9.46e5·14-s + 1.20e5·15-s + 1.49e7·16-s + 7.19e6·17-s − 5.13e6·18-s + 6.26e6·19-s − 2.74e6·20-s − 2.64e6·21-s − 2.70e7·22-s + 1.54e7·23-s + 7.34e7·24-s − 4.85e7·25-s + 9.56e7·26-s − 1.43e7·27-s + 6.00e7·28-s + ⋯
L(s)  = 1  − 1.92·2-s − 0.577·3-s + 2.69·4-s − 0.0712·5-s + 1.10·6-s + 0.244·7-s − 3.26·8-s + 0.333·9-s + 0.136·10-s + 0.582·11-s − 1.55·12-s − 0.820·13-s − 0.470·14-s + 0.0411·15-s + 3.57·16-s + 1.22·17-s − 0.640·18-s + 0.580·19-s − 0.192·20-s − 0.141·21-s − 1.11·22-s + 0.500·23-s + 1.88·24-s − 0.994·25-s + 1.57·26-s − 0.192·27-s + 0.659·28-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 \( 1 + 87.0T + 2.04e3T^{2} \)
5 \( 1 + 497.T + 4.88e7T^{2} \)
7 \( 1 - 1.08e4T + 1.97e9T^{2} \)
11 \( 1 - 3.10e5T + 2.85e11T^{2} \)
13 \( 1 + 1.09e6T + 1.79e12T^{2} \)
17 \( 1 - 7.19e6T + 3.42e13T^{2} \)
19 \( 1 - 6.26e6T + 1.16e14T^{2} \)
23 \( 1 - 1.54e7T + 9.52e14T^{2} \)
29 \( 1 + 7.10e6T + 1.22e16T^{2} \)
31 \( 1 - 4.03e7T + 2.54e16T^{2} \)
37 \( 1 + 7.18e8T + 1.77e17T^{2} \)
41 \( 1 - 3.85e8T + 5.50e17T^{2} \)
43 \( 1 + 6.53e8T + 9.29e17T^{2} \)
47 \( 1 - 5.83e8T + 2.47e18T^{2} \)
53 \( 1 + 4.97e9T + 9.26e18T^{2} \)
61 \( 1 - 4.77e8T + 4.35e19T^{2} \)
67 \( 1 + 2.18e9T + 1.22e20T^{2} \)
71 \( 1 - 1.54e10T + 2.31e20T^{2} \)
73 \( 1 - 2.19e10T + 3.13e20T^{2} \)
79 \( 1 + 4.35e9T + 7.47e20T^{2} \)
83 \( 1 - 6.40e10T + 1.28e21T^{2} \)
89 \( 1 + 4.53e10T + 2.77e21T^{2} \)
97 \( 1 + 1.41e11T + 7.15e21T^{2} \)
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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

−9.944499213947863726846447055624, −9.381771055854299665144317685159, −8.137408525010591574234937780786, −7.38877669361174712648824407808, −6.47092140261471750896643654307, −5.28307460190012683297237861513, −3.31197165790883757307915572532, −1.89626068026268920410209236313, −1.01291671806365823166109879874, 0, 1.01291671806365823166109879874, 1.89626068026268920410209236313, 3.31197165790883757307915572532, 5.28307460190012683297237861513, 6.47092140261471750896643654307, 7.38877669361174712648824407808, 8.137408525010591574234937780786, 9.381771055854299665144317685159, 9.944499213947863726846447055624

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