Properties

Label 2-177-1.1-c11-0-104
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  + 76.3·2-s − 243·3-s + 3.78e3·4-s + 5.26e3·5-s − 1.85e4·6-s + 2.13e4·7-s + 1.32e5·8-s + 5.90e4·9-s + 4.01e5·10-s − 3.98e5·11-s − 9.18e5·12-s − 5.98e5·13-s + 1.62e6·14-s − 1.27e6·15-s + 2.35e6·16-s − 9.96e6·17-s + 4.50e6·18-s − 5.59e6·19-s + 1.98e7·20-s − 5.18e6·21-s − 3.04e7·22-s − 1.67e7·23-s − 3.21e7·24-s − 2.11e7·25-s − 4.57e7·26-s − 1.43e7·27-s + 8.05e7·28-s + ⋯
L(s)  = 1  + 1.68·2-s − 0.577·3-s + 1.84·4-s + 0.752·5-s − 0.974·6-s + 0.479·7-s + 1.42·8-s + 0.333·9-s + 1.27·10-s − 0.746·11-s − 1.06·12-s − 0.447·13-s + 0.808·14-s − 0.434·15-s + 0.561·16-s − 1.70·17-s + 0.562·18-s − 0.518·19-s + 1.39·20-s − 0.276·21-s − 1.25·22-s − 0.542·23-s − 0.824·24-s − 0.433·25-s − 0.754·26-s − 0.192·27-s + 0.885·28-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 \( 1 - 76.3T + 2.04e3T^{2} \)
5 \( 1 - 5.26e3T + 4.88e7T^{2} \)
7 \( 1 - 2.13e4T + 1.97e9T^{2} \)
11 \( 1 + 3.98e5T + 2.85e11T^{2} \)
13 \( 1 + 5.98e5T + 1.79e12T^{2} \)
17 \( 1 + 9.96e6T + 3.42e13T^{2} \)
19 \( 1 + 5.59e6T + 1.16e14T^{2} \)
23 \( 1 + 1.67e7T + 9.52e14T^{2} \)
29 \( 1 + 8.80e7T + 1.22e16T^{2} \)
31 \( 1 - 5.31e7T + 2.54e16T^{2} \)
37 \( 1 - 2.54e8T + 1.77e17T^{2} \)
41 \( 1 - 1.12e9T + 5.50e17T^{2} \)
43 \( 1 + 7.10e7T + 9.29e17T^{2} \)
47 \( 1 - 1.50e9T + 2.47e18T^{2} \)
53 \( 1 + 5.65e9T + 9.26e18T^{2} \)
61 \( 1 - 1.97e9T + 4.35e19T^{2} \)
67 \( 1 + 1.70e10T + 1.22e20T^{2} \)
71 \( 1 + 1.45e10T + 2.31e20T^{2} \)
73 \( 1 - 1.40e10T + 3.13e20T^{2} \)
79 \( 1 - 4.37e10T + 7.47e20T^{2} \)
83 \( 1 - 1.87e10T + 1.28e21T^{2} \)
89 \( 1 - 5.54e10T + 2.77e21T^{2} \)
97 \( 1 + 1.57e11T + 7.15e21T^{2} \)
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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

−10.69551219936795671485366101097, −9.358635768490403029752822724357, −7.72796700482015958471985738292, −6.51007123307516007048553349645, −5.81645189404258429236819965428, −4.87588441004005852911110920883, −4.16063798607595002822066813191, −2.56833831714961643591436991959, −1.86888838452920132534993653762, 0, 1.86888838452920132534993653762, 2.56833831714961643591436991959, 4.16063798607595002822066813191, 4.87588441004005852911110920883, 5.81645189404258429236819965428, 6.51007123307516007048553349645, 7.72796700482015958471985738292, 9.358635768490403029752822724357, 10.69551219936795671485366101097

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