Properties

Label 2-177-1.1-c13-0-45
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 + 2.03e4·4-s + 1.50e4·5-s + 1.23e5·6-s − 2.57e5·7-s − 2.06e6·8-s + 5.31e5·9-s − 2.54e6·10-s − 4.81e6·11-s − 1.48e7·12-s − 3.34e7·13-s + 4.35e7·14-s − 1.09e7·15-s + 1.81e8·16-s − 1.05e8·17-s − 8.98e7·18-s + 1.32e8·19-s + 3.06e8·20-s + 1.87e8·21-s + 8.13e8·22-s + 7.80e8·23-s + 1.50e9·24-s − 9.94e8·25-s + 5.66e9·26-s − 3.87e8·27-s − 5.25e9·28-s + ⋯
L(s)  = 1  − 1.86·2-s − 0.577·3-s + 2.48·4-s + 0.430·5-s + 1.07·6-s − 0.826·7-s − 2.78·8-s + 0.333·9-s − 0.803·10-s − 0.819·11-s − 1.43·12-s − 1.92·13-s + 1.54·14-s − 0.248·15-s + 2.70·16-s − 1.06·17-s − 0.622·18-s + 0.646·19-s + 1.07·20-s + 0.477·21-s + 1.53·22-s + 1.09·23-s + 1.60·24-s − 0.814·25-s + 3.59·26-s − 0.192·27-s − 2.05·28-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 \( 1 - 1.50e4T + 1.22e9T^{2} \)
7 \( 1 + 2.57e5T + 9.68e10T^{2} \)
11 \( 1 + 4.81e6T + 3.45e13T^{2} \)
13 \( 1 + 3.34e7T + 3.02e14T^{2} \)
17 \( 1 + 1.05e8T + 9.90e15T^{2} \)
19 \( 1 - 1.32e8T + 4.20e16T^{2} \)
23 \( 1 - 7.80e8T + 5.04e17T^{2} \)
29 \( 1 + 9.46e8T + 1.02e19T^{2} \)
31 \( 1 - 7.66e9T + 2.44e19T^{2} \)
37 \( 1 - 1.25e10T + 2.43e20T^{2} \)
41 \( 1 + 2.31e10T + 9.25e20T^{2} \)
43 \( 1 - 2.54e10T + 1.71e21T^{2} \)
47 \( 1 - 9.37e9T + 5.46e21T^{2} \)
53 \( 1 - 1.13e11T + 2.60e22T^{2} \)
61 \( 1 + 1.44e11T + 1.61e23T^{2} \)
67 \( 1 + 4.28e11T + 5.48e23T^{2} \)
71 \( 1 - 1.96e12T + 1.16e24T^{2} \)
73 \( 1 + 1.02e12T + 1.67e24T^{2} \)
79 \( 1 - 7.60e11T + 4.66e24T^{2} \)
83 \( 1 - 1.79e12T + 8.87e24T^{2} \)
89 \( 1 - 8.56e12T + 2.19e25T^{2} \)
97 \( 1 - 1.76e12T + 6.73e25T^{2} \)
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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

−9.808867060922721098769630364897, −9.179319971378387491470413223141, −7.82962808747132586691243678991, −7.06571697294410024055227162372, −6.22861708446718597567196205642, −4.97484125373384975026052213632, −2.83020384061588135075823955406, −2.13936529257741338964172842554, −0.73579023098383782971654199549, 0, 0.73579023098383782971654199549, 2.13936529257741338964172842554, 2.83020384061588135075823955406, 4.97484125373384975026052213632, 6.22861708446718597567196205642, 7.06571697294410024055227162372, 7.82962808747132586691243678991, 9.179319971378387491470413223141, 9.808867060922721098769630364897

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