Properties

Label 2-177-1.1-c13-0-80
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  − 169.·2-s + 729·3-s + 2.05e4·4-s − 2.14e4·5-s − 1.23e5·6-s + 3.61e5·7-s − 2.09e6·8-s + 5.31e5·9-s + 3.63e6·10-s − 6.55e6·11-s + 1.49e7·12-s + 3.19e6·13-s − 6.12e7·14-s − 1.56e7·15-s + 1.86e8·16-s + 2.70e7·17-s − 9.00e7·18-s − 6.84e7·19-s − 4.40e8·20-s + 2.63e8·21-s + 1.11e9·22-s − 5.78e8·23-s − 1.52e9·24-s − 7.62e8·25-s − 5.41e8·26-s + 3.87e8·27-s + 7.42e9·28-s + ⋯
L(s)  = 1  − 1.87·2-s + 0.577·3-s + 2.50·4-s − 0.612·5-s − 1.08·6-s + 1.16·7-s − 2.82·8-s + 0.333·9-s + 1.14·10-s − 1.11·11-s + 1.44·12-s + 0.183·13-s − 2.17·14-s − 0.353·15-s + 2.78·16-s + 0.271·17-s − 0.624·18-s − 0.333·19-s − 1.53·20-s + 0.669·21-s + 2.09·22-s − 0.814·23-s − 1.63·24-s − 0.624·25-s − 0.343·26-s + 0.192·27-s + 2.91·28-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 \( 1 + 169.T + 8.19e3T^{2} \)
5 \( 1 + 2.14e4T + 1.22e9T^{2} \)
7 \( 1 - 3.61e5T + 9.68e10T^{2} \)
11 \( 1 + 6.55e6T + 3.45e13T^{2} \)
13 \( 1 - 3.19e6T + 3.02e14T^{2} \)
17 \( 1 - 2.70e7T + 9.90e15T^{2} \)
19 \( 1 + 6.84e7T + 4.20e16T^{2} \)
23 \( 1 + 5.78e8T + 5.04e17T^{2} \)
29 \( 1 - 3.44e9T + 1.02e19T^{2} \)
31 \( 1 - 9.42e9T + 2.44e19T^{2} \)
37 \( 1 - 1.03e10T + 2.43e20T^{2} \)
41 \( 1 + 2.19e9T + 9.25e20T^{2} \)
43 \( 1 + 2.12e10T + 1.71e21T^{2} \)
47 \( 1 - 3.14e10T + 5.46e21T^{2} \)
53 \( 1 + 2.87e11T + 2.60e22T^{2} \)
61 \( 1 + 5.01e10T + 1.61e23T^{2} \)
67 \( 1 + 5.06e11T + 5.48e23T^{2} \)
71 \( 1 + 1.47e12T + 1.16e24T^{2} \)
73 \( 1 + 5.72e11T + 1.67e24T^{2} \)
79 \( 1 - 1.61e12T + 4.66e24T^{2} \)
83 \( 1 - 2.15e12T + 8.87e24T^{2} \)
89 \( 1 - 1.78e12T + 2.19e25T^{2} \)
97 \( 1 + 3.76e11T + 6.73e25T^{2} \)
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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

−9.824880728473388617280384860990, −8.522821244557925788960448305998, −8.058831836070132030359995308830, −7.56455680017789649546532057595, −6.22204597929045235691242169963, −4.59142618438619139425629483060, −2.96582648477584626514824797010, −2.04412654905221836736917116190, −1.07258993715244843308453709531, 0, 1.07258993715244843308453709531, 2.04412654905221836736917116190, 2.96582648477584626514824797010, 4.59142618438619139425629483060, 6.22204597929045235691242169963, 7.56455680017789649546532057595, 8.058831836070132030359995308830, 8.522821244557925788960448305998, 9.824880728473388617280384860990

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