Properties

Label 2-177-1.1-c13-0-81
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  − 117.·2-s − 729·3-s + 5.67e3·4-s − 2.82e4·5-s + 8.58e4·6-s + 2.97e5·7-s + 2.96e5·8-s + 5.31e5·9-s + 3.33e6·10-s + 6.74e6·11-s − 4.13e6·12-s + 1.09e7·13-s − 3.50e7·14-s + 2.06e7·15-s − 8.13e7·16-s + 1.25e8·17-s − 6.25e7·18-s + 5.93e7·19-s − 1.60e8·20-s − 2.16e8·21-s − 7.93e8·22-s − 2.24e8·23-s − 2.16e8·24-s − 4.19e8·25-s − 1.29e9·26-s − 3.87e8·27-s + 1.68e9·28-s + ⋯
L(s)  = 1  − 1.30·2-s − 0.577·3-s + 0.692·4-s − 0.809·5-s + 0.751·6-s + 0.956·7-s + 0.399·8-s + 0.333·9-s + 1.05·10-s + 1.14·11-s − 0.400·12-s + 0.629·13-s − 1.24·14-s + 0.467·15-s − 1.21·16-s + 1.25·17-s − 0.433·18-s + 0.289·19-s − 0.561·20-s − 0.551·21-s − 1.49·22-s − 0.316·23-s − 0.230·24-s − 0.344·25-s − 0.819·26-s − 0.192·27-s + 0.662·28-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 \( 1 + 117.T + 8.19e3T^{2} \)
5 \( 1 + 2.82e4T + 1.22e9T^{2} \)
7 \( 1 - 2.97e5T + 9.68e10T^{2} \)
11 \( 1 - 6.74e6T + 3.45e13T^{2} \)
13 \( 1 - 1.09e7T + 3.02e14T^{2} \)
17 \( 1 - 1.25e8T + 9.90e15T^{2} \)
19 \( 1 - 5.93e7T + 4.20e16T^{2} \)
23 \( 1 + 2.24e8T + 5.04e17T^{2} \)
29 \( 1 + 3.59e9T + 1.02e19T^{2} \)
31 \( 1 - 2.51e9T + 2.44e19T^{2} \)
37 \( 1 + 9.31e9T + 2.43e20T^{2} \)
41 \( 1 + 2.88e10T + 9.25e20T^{2} \)
43 \( 1 - 6.45e10T + 1.71e21T^{2} \)
47 \( 1 + 3.47e9T + 5.46e21T^{2} \)
53 \( 1 + 6.81e10T + 2.60e22T^{2} \)
61 \( 1 + 2.72e11T + 1.61e23T^{2} \)
67 \( 1 + 1.27e12T + 5.48e23T^{2} \)
71 \( 1 - 2.97e11T + 1.16e24T^{2} \)
73 \( 1 + 7.41e11T + 1.67e24T^{2} \)
79 \( 1 + 2.93e12T + 4.66e24T^{2} \)
83 \( 1 - 3.27e12T + 8.87e24T^{2} \)
89 \( 1 - 1.53e12T + 2.19e25T^{2} \)
97 \( 1 + 6.38e12T + 6.73e25T^{2} \)
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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

−9.784511922671488409093606699353, −8.804123261538606498711047158715, −7.901849677056997889970876227864, −7.26830868578345688556103738328, −5.92606042369651611373412821331, −4.60229779141439341985405176571, −3.64612311399667995087285110319, −1.65330329297034497980912185343, −1.07418361251261334468139524503, 0, 1.07418361251261334468139524503, 1.65330329297034497980912185343, 3.64612311399667995087285110319, 4.60229779141439341985405176571, 5.92606042369651611373412821331, 7.26830868578345688556103738328, 7.901849677056997889970876227864, 8.804123261538606498711047158715, 9.784511922671488409093606699353

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