Properties

Label 2-177-1.1-c11-0-43
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  + 27.2·2-s − 243·3-s − 1.30e3·4-s − 8.75e3·5-s − 6.61e3·6-s − 6.82e4·7-s − 9.13e4·8-s + 5.90e4·9-s − 2.38e5·10-s + 6.64e5·11-s + 3.17e5·12-s − 1.65e6·13-s − 1.85e6·14-s + 2.12e6·15-s + 1.90e5·16-s + 1.11e7·17-s + 1.60e6·18-s − 8.29e5·19-s + 1.14e7·20-s + 1.65e7·21-s + 1.80e7·22-s + 1.30e6·23-s + 2.21e7·24-s + 2.77e7·25-s − 4.50e7·26-s − 1.43e7·27-s + 8.91e7·28-s + ⋯
L(s)  = 1  + 0.601·2-s − 0.577·3-s − 0.638·4-s − 1.25·5-s − 0.347·6-s − 1.53·7-s − 0.985·8-s + 0.333·9-s − 0.753·10-s + 1.24·11-s + 0.368·12-s − 1.23·13-s − 0.922·14-s + 0.722·15-s + 0.0454·16-s + 1.90·17-s + 0.200·18-s − 0.0768·19-s + 0.799·20-s + 0.885·21-s + 0.747·22-s + 0.0422·23-s + 0.568·24-s + 0.568·25-s − 0.743·26-s − 0.192·27-s + 0.979·28-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 \( 1 - 27.2T + 2.04e3T^{2} \)
5 \( 1 + 8.75e3T + 4.88e7T^{2} \)
7 \( 1 + 6.82e4T + 1.97e9T^{2} \)
11 \( 1 - 6.64e5T + 2.85e11T^{2} \)
13 \( 1 + 1.65e6T + 1.79e12T^{2} \)
17 \( 1 - 1.11e7T + 3.42e13T^{2} \)
19 \( 1 + 8.29e5T + 1.16e14T^{2} \)
23 \( 1 - 1.30e6T + 9.52e14T^{2} \)
29 \( 1 + 4.39e7T + 1.22e16T^{2} \)
31 \( 1 - 1.84e8T + 2.54e16T^{2} \)
37 \( 1 - 7.52e7T + 1.77e17T^{2} \)
41 \( 1 + 5.89e8T + 5.50e17T^{2} \)
43 \( 1 - 1.11e9T + 9.29e17T^{2} \)
47 \( 1 + 2.06e9T + 2.47e18T^{2} \)
53 \( 1 + 3.47e9T + 9.26e18T^{2} \)
61 \( 1 + 2.03e9T + 4.35e19T^{2} \)
67 \( 1 + 3.18e9T + 1.22e20T^{2} \)
71 \( 1 - 1.90e10T + 2.31e20T^{2} \)
73 \( 1 - 2.41e10T + 3.13e20T^{2} \)
79 \( 1 - 1.77e10T + 7.47e20T^{2} \)
83 \( 1 + 3.46e10T + 1.28e21T^{2} \)
89 \( 1 + 6.60e10T + 2.77e21T^{2} \)
97 \( 1 - 4.44e10T + 7.15e21T^{2} \)
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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

−9.938101755376230997907519184390, −9.459168950387966417622699632209, −8.016280771268722345531168280998, −6.89708143654857606271358583594, −5.94146991010975934186396559243, −4.75224592371613339839789374911, −3.75721846454186107576946757452, −3.14680488394737555485484591828, −0.820125545011874894945188585694, 0, 0.820125545011874894945188585694, 3.14680488394737555485484591828, 3.75721846454186107576946757452, 4.75224592371613339839789374911, 5.94146991010975934186396559243, 6.89708143654857606271358583594, 8.016280771268722345531168280998, 9.459168950387966417622699632209, 9.938101755376230997907519184390

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