Properties

Label 2-177-1.1-c11-0-99
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  + 57.7·2-s + 243·3-s + 1.28e3·4-s + 767.·5-s + 1.40e4·6-s − 3.13e4·7-s − 4.40e4·8-s + 5.90e4·9-s + 4.43e4·10-s + 6.11e5·11-s + 3.12e5·12-s − 1.76e5·13-s − 1.80e6·14-s + 1.86e5·15-s − 5.17e6·16-s − 4.51e5·17-s + 3.40e6·18-s + 5.30e6·19-s + 9.87e5·20-s − 7.60e6·21-s + 3.53e7·22-s − 4.82e7·23-s − 1.06e7·24-s − 4.82e7·25-s − 1.02e7·26-s + 1.43e7·27-s − 4.02e7·28-s + ⋯
L(s)  = 1  + 1.27·2-s + 0.577·3-s + 0.627·4-s + 0.109·5-s + 0.736·6-s − 0.704·7-s − 0.474·8-s + 0.333·9-s + 0.140·10-s + 1.14·11-s + 0.362·12-s − 0.132·13-s − 0.898·14-s + 0.0634·15-s − 1.23·16-s − 0.0771·17-s + 0.425·18-s + 0.491·19-s + 0.0689·20-s − 0.406·21-s + 1.46·22-s − 1.56·23-s − 0.274·24-s − 0.987·25-s − 0.168·26-s + 0.192·27-s − 0.441·28-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 \( 1 - 57.7T + 2.04e3T^{2} \)
5 \( 1 - 767.T + 4.88e7T^{2} \)
7 \( 1 + 3.13e4T + 1.97e9T^{2} \)
11 \( 1 - 6.11e5T + 2.85e11T^{2} \)
13 \( 1 + 1.76e5T + 1.79e12T^{2} \)
17 \( 1 + 4.51e5T + 3.42e13T^{2} \)
19 \( 1 - 5.30e6T + 1.16e14T^{2} \)
23 \( 1 + 4.82e7T + 9.52e14T^{2} \)
29 \( 1 - 6.91e7T + 1.22e16T^{2} \)
31 \( 1 - 1.50e8T + 2.54e16T^{2} \)
37 \( 1 + 1.05e8T + 1.77e17T^{2} \)
41 \( 1 - 2.19e8T + 5.50e17T^{2} \)
43 \( 1 + 1.08e9T + 9.29e17T^{2} \)
47 \( 1 + 2.42e9T + 2.47e18T^{2} \)
53 \( 1 + 1.46e9T + 9.26e18T^{2} \)
61 \( 1 + 1.10e10T + 4.35e19T^{2} \)
67 \( 1 + 4.40e9T + 1.22e20T^{2} \)
71 \( 1 + 5.15e9T + 2.31e20T^{2} \)
73 \( 1 - 1.29e10T + 3.13e20T^{2} \)
79 \( 1 + 1.77e10T + 7.47e20T^{2} \)
83 \( 1 + 3.33e10T + 1.28e21T^{2} \)
89 \( 1 - 4.53e10T + 2.77e21T^{2} \)
97 \( 1 + 8.62e10T + 7.15e21T^{2} \)
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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

−9.990317116879671230323739638047, −9.302626639611724106559137139328, −8.104860657808942985555702739663, −6.66445452066083878247542772139, −6.02105052888455048444908820531, −4.64545885536203076213264119256, −3.75400216086156235931412011088, −2.96126343695728614646464692192, −1.68378125483218692937788277298, 0, 1.68378125483218692937788277298, 2.96126343695728614646464692192, 3.75400216086156235931412011088, 4.64545885536203076213264119256, 6.02105052888455048444908820531, 6.66445452066083878247542772139, 8.104860657808942985555702739663, 9.302626639611724106559137139328, 9.990317116879671230323739638047

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