Properties

Label 2-177-1.1-c13-0-122
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 165.·2-s + 729·3-s + 1.92e4·4-s − 4.53e4·5-s + 1.20e5·6-s + 5.05e5·7-s + 1.83e6·8-s + 5.31e5·9-s − 7.52e6·10-s − 4.65e6·11-s + 1.40e7·12-s − 3.25e7·13-s + 8.37e7·14-s − 3.30e7·15-s + 1.46e8·16-s − 8.81e7·17-s + 8.80e7·18-s − 2.09e8·19-s − 8.75e8·20-s + 3.68e8·21-s − 7.72e8·22-s − 1.11e9·23-s + 1.34e9·24-s + 8.39e8·25-s − 5.39e9·26-s + 3.87e8·27-s + 9.74e9·28-s + ⋯
L(s)  = 1  + 1.83·2-s + 0.577·3-s + 2.35·4-s − 1.29·5-s + 1.05·6-s + 1.62·7-s + 2.48·8-s + 0.333·9-s − 2.37·10-s − 0.792·11-s + 1.35·12-s − 1.86·13-s + 2.97·14-s − 0.750·15-s + 2.18·16-s − 0.886·17-s + 0.610·18-s − 1.02·19-s − 3.05·20-s + 0.937·21-s − 1.45·22-s − 1.57·23-s + 1.43·24-s + 0.687·25-s − 3.42·26-s + 0.192·27-s + 3.82·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 729T \)
59 \( 1 - 4.21e10T \)
good2 \( 1 - 165.T + 8.19e3T^{2} \)
5 \( 1 + 4.53e4T + 1.22e9T^{2} \)
7 \( 1 - 5.05e5T + 9.68e10T^{2} \)
11 \( 1 + 4.65e6T + 3.45e13T^{2} \)
13 \( 1 + 3.25e7T + 3.02e14T^{2} \)
17 \( 1 + 8.81e7T + 9.90e15T^{2} \)
19 \( 1 + 2.09e8T + 4.20e16T^{2} \)
23 \( 1 + 1.11e9T + 5.04e17T^{2} \)
29 \( 1 - 4.51e8T + 1.02e19T^{2} \)
31 \( 1 - 3.88e9T + 2.44e19T^{2} \)
37 \( 1 + 4.42e8T + 2.43e20T^{2} \)
41 \( 1 + 4.71e10T + 9.25e20T^{2} \)
43 \( 1 - 2.64e10T + 1.71e21T^{2} \)
47 \( 1 + 6.94e10T + 5.46e21T^{2} \)
53 \( 1 - 1.06e11T + 2.60e22T^{2} \)
61 \( 1 + 6.28e10T + 1.61e23T^{2} \)
67 \( 1 - 1.93e11T + 5.48e23T^{2} \)
71 \( 1 - 1.59e12T + 1.16e24T^{2} \)
73 \( 1 - 1.32e12T + 1.67e24T^{2} \)
79 \( 1 - 2.93e12T + 4.66e24T^{2} \)
83 \( 1 + 5.25e12T + 8.87e24T^{2} \)
89 \( 1 - 1.71e12T + 2.19e25T^{2} \)
97 \( 1 + 7.81e12T + 6.73e25T^{2} \)
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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

−10.31582493130136406842491187348, −8.219768093516591033449970548827, −7.76743265690420860665381277101, −6.78516015622285592079326776450, −5.14262804483160542012805152445, −4.57015150338992377263450056879, −3.91071280566062324347050834687, −2.53059515952652912761465903549, −1.98591232812066276563785643596, 0, 1.98591232812066276563785643596, 2.53059515952652912761465903549, 3.91071280566062324347050834687, 4.57015150338992377263450056879, 5.14262804483160542012805152445, 6.78516015622285592079326776450, 7.76743265690420860665381277101, 8.219768093516591033449970548827, 10.31582493130136406842491187348

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