Properties

Label 2-177-1.1-c11-0-93
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 35.0·2-s + 243·3-s − 820.·4-s + 1.50e3·5-s + 8.51e3·6-s + 260.·7-s − 1.00e5·8-s + 5.90e4·9-s + 5.28e4·10-s − 3.36e5·11-s − 1.99e5·12-s − 1.03e6·13-s + 9.12e3·14-s + 3.66e5·15-s − 1.83e6·16-s + 1.09e7·17-s + 2.06e6·18-s + 1.71e7·19-s − 1.23e6·20-s + 6.33e4·21-s − 1.17e7·22-s + 5.60e7·23-s − 2.44e7·24-s − 4.65e7·25-s − 3.63e7·26-s + 1.43e7·27-s − 2.13e5·28-s + ⋯
L(s)  = 1  + 0.774·2-s + 0.577·3-s − 0.400·4-s + 0.216·5-s + 0.446·6-s + 0.00585·7-s − 1.08·8-s + 0.333·9-s + 0.167·10-s − 0.629·11-s − 0.231·12-s − 0.774·13-s + 0.00453·14-s + 0.124·15-s − 0.438·16-s + 1.86·17-s + 0.258·18-s + 1.58·19-s − 0.0865·20-s + 0.00338·21-s − 0.487·22-s + 1.81·23-s − 0.626·24-s − 0.953·25-s − 0.599·26-s + 0.192·27-s − 0.00234·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 243T \)
59 \( 1 + 7.14e8T \)
good2 \( 1 - 35.0T + 2.04e3T^{2} \)
5 \( 1 - 1.50e3T + 4.88e7T^{2} \)
7 \( 1 - 260.T + 1.97e9T^{2} \)
11 \( 1 + 3.36e5T + 2.85e11T^{2} \)
13 \( 1 + 1.03e6T + 1.79e12T^{2} \)
17 \( 1 - 1.09e7T + 3.42e13T^{2} \)
19 \( 1 - 1.71e7T + 1.16e14T^{2} \)
23 \( 1 - 5.60e7T + 9.52e14T^{2} \)
29 \( 1 + 1.56e8T + 1.22e16T^{2} \)
31 \( 1 + 1.86e8T + 2.54e16T^{2} \)
37 \( 1 + 3.36e8T + 1.77e17T^{2} \)
41 \( 1 + 7.88e7T + 5.50e17T^{2} \)
43 \( 1 + 6.26e8T + 9.29e17T^{2} \)
47 \( 1 + 1.23e9T + 2.47e18T^{2} \)
53 \( 1 + 4.49e9T + 9.26e18T^{2} \)
61 \( 1 + 2.43e9T + 4.35e19T^{2} \)
67 \( 1 - 1.79e10T + 1.22e20T^{2} \)
71 \( 1 - 2.14e10T + 2.31e20T^{2} \)
73 \( 1 + 2.60e10T + 3.13e20T^{2} \)
79 \( 1 + 1.34e9T + 7.47e20T^{2} \)
83 \( 1 + 4.68e10T + 1.28e21T^{2} \)
89 \( 1 + 4.74e10T + 2.77e21T^{2} \)
97 \( 1 + 1.01e11T + 7.15e21T^{2} \)
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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

−9.800461323199176344384087513724, −9.425283794284371785143082970513, −8.044033876162937678164026971659, −7.19575823655854083019660491809, −5.48849968605529754179857787577, −5.10028704761213134554825327327, −3.53728947325539358608559054863, −2.98805994357525852707149853351, −1.44649847319511324396052090724, 0, 1.44649847319511324396052090724, 2.98805994357525852707149853351, 3.53728947325539358608559054863, 5.10028704761213134554825327327, 5.48849968605529754179857787577, 7.19575823655854083019660491809, 8.044033876162937678164026971659, 9.425283794284371785143082970513, 9.800461323199176344384087513724

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