Properties

Label 2-177-1.1-c13-0-78
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  − 17.8·2-s + 729·3-s − 7.87e3·4-s + 5.82e3·5-s − 1.29e4·6-s − 4.95e5·7-s + 2.86e5·8-s + 5.31e5·9-s − 1.03e5·10-s + 4.50e6·11-s − 5.74e6·12-s + 2.28e7·13-s + 8.83e6·14-s + 4.24e6·15-s + 5.94e7·16-s − 1.25e8·17-s − 9.46e6·18-s − 1.03e8·19-s − 4.58e7·20-s − 3.61e8·21-s − 8.02e7·22-s − 1.20e9·23-s + 2.08e8·24-s − 1.18e9·25-s − 4.06e8·26-s + 3.87e8·27-s + 3.90e9·28-s + ⋯
L(s)  = 1  − 0.196·2-s + 0.577·3-s − 0.961·4-s + 0.166·5-s − 0.113·6-s − 1.59·7-s + 0.386·8-s + 0.333·9-s − 0.0328·10-s + 0.766·11-s − 0.554·12-s + 1.31·13-s + 0.313·14-s + 0.0962·15-s + 0.885·16-s − 1.26·17-s − 0.0656·18-s − 0.502·19-s − 0.160·20-s − 0.919·21-s − 0.150·22-s − 1.69·23-s + 0.222·24-s − 0.972·25-s − 0.258·26-s + 0.192·27-s + 1.53·28-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 \( 1 + 17.8T + 8.19e3T^{2} \)
5 \( 1 - 5.82e3T + 1.22e9T^{2} \)
7 \( 1 + 4.95e5T + 9.68e10T^{2} \)
11 \( 1 - 4.50e6T + 3.45e13T^{2} \)
13 \( 1 - 2.28e7T + 3.02e14T^{2} \)
17 \( 1 + 1.25e8T + 9.90e15T^{2} \)
19 \( 1 + 1.03e8T + 4.20e16T^{2} \)
23 \( 1 + 1.20e9T + 5.04e17T^{2} \)
29 \( 1 - 4.62e9T + 1.02e19T^{2} \)
31 \( 1 - 8.13e9T + 2.44e19T^{2} \)
37 \( 1 + 7.22e9T + 2.43e20T^{2} \)
41 \( 1 - 9.28e9T + 9.25e20T^{2} \)
43 \( 1 - 3.44e10T + 1.71e21T^{2} \)
47 \( 1 - 3.11e10T + 5.46e21T^{2} \)
53 \( 1 - 6.65e10T + 2.60e22T^{2} \)
61 \( 1 - 6.87e11T + 1.61e23T^{2} \)
67 \( 1 - 5.94e11T + 5.48e23T^{2} \)
71 \( 1 - 6.97e11T + 1.16e24T^{2} \)
73 \( 1 + 1.61e12T + 1.67e24T^{2} \)
79 \( 1 + 3.41e12T + 4.66e24T^{2} \)
83 \( 1 - 2.41e12T + 8.87e24T^{2} \)
89 \( 1 - 3.97e12T + 2.19e25T^{2} \)
97 \( 1 + 7.61e12T + 6.73e25T^{2} \)
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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

−9.728413875707428318827516211574, −8.867823869658786574950350008804, −8.244041656751976518329779494331, −6.63672002545926155151470177055, −6.02134626522299491179480568109, −4.20798055304486458861825545710, −3.75746890633151087727274663250, −2.45036353200641753548425084321, −1.04088980173912984193451618567, 0, 1.04088980173912984193451618567, 2.45036353200641753548425084321, 3.75746890633151087727274663250, 4.20798055304486458861825545710, 6.02134626522299491179480568109, 6.63672002545926155151470177055, 8.244041656751976518329779494331, 8.867823869658786574950350008804, 9.728413875707428318827516211574

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