Properties

Label 2-177-1.1-c11-0-89
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  + 10.6·2-s + 243·3-s − 1.93e3·4-s + 1.26e4·5-s + 2.59e3·6-s − 7.56e4·7-s − 4.25e4·8-s + 5.90e4·9-s + 1.34e5·10-s + 7.75e4·11-s − 4.69e5·12-s + 1.47e6·13-s − 8.09e5·14-s + 3.06e6·15-s + 3.50e6·16-s − 9.88e6·17-s + 6.31e5·18-s − 5.07e6·19-s − 2.43e7·20-s − 1.83e7·21-s + 8.29e5·22-s + 4.46e7·23-s − 1.03e7·24-s + 1.10e8·25-s + 1.58e7·26-s + 1.43e7·27-s + 1.46e8·28-s + ⋯
L(s)  = 1  + 0.236·2-s + 0.577·3-s − 0.944·4-s + 1.80·5-s + 0.136·6-s − 1.70·7-s − 0.459·8-s + 0.333·9-s + 0.426·10-s + 0.145·11-s − 0.545·12-s + 1.10·13-s − 0.402·14-s + 1.04·15-s + 0.835·16-s − 1.68·17-s + 0.0787·18-s − 0.469·19-s − 1.70·20-s − 0.982·21-s + 0.0342·22-s + 1.44·23-s − 0.265·24-s + 2.25·25-s + 0.261·26-s + 0.192·27-s + 1.60·28-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 \( 1 - 10.6T + 2.04e3T^{2} \)
5 \( 1 - 1.26e4T + 4.88e7T^{2} \)
7 \( 1 + 7.56e4T + 1.97e9T^{2} \)
11 \( 1 - 7.75e4T + 2.85e11T^{2} \)
13 \( 1 - 1.47e6T + 1.79e12T^{2} \)
17 \( 1 + 9.88e6T + 3.42e13T^{2} \)
19 \( 1 + 5.07e6T + 1.16e14T^{2} \)
23 \( 1 - 4.46e7T + 9.52e14T^{2} \)
29 \( 1 + 8.95e7T + 1.22e16T^{2} \)
31 \( 1 - 4.40e7T + 2.54e16T^{2} \)
37 \( 1 + 3.52e7T + 1.77e17T^{2} \)
41 \( 1 - 5.31e8T + 5.50e17T^{2} \)
43 \( 1 + 1.39e9T + 9.29e17T^{2} \)
47 \( 1 + 2.26e9T + 2.47e18T^{2} \)
53 \( 1 + 5.83e9T + 9.26e18T^{2} \)
61 \( 1 - 1.97e9T + 4.35e19T^{2} \)
67 \( 1 + 1.49e10T + 1.22e20T^{2} \)
71 \( 1 - 1.84e10T + 2.31e20T^{2} \)
73 \( 1 - 1.03e10T + 3.13e20T^{2} \)
79 \( 1 + 4.76e9T + 7.47e20T^{2} \)
83 \( 1 + 6.60e10T + 1.28e21T^{2} \)
89 \( 1 - 6.92e10T + 2.77e21T^{2} \)
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

−9.790324362389914158199638704682, −9.229843429834738017743771727697, −8.696112485531838131021072172633, −6.60857182689548228374724590554, −6.19128254987258743371122786314, −4.91838388619518916977402142883, −3.58223164072139175138986386344, −2.68297506313966791050026815080, −1.40825868741763675729884159703, 0, 1.40825868741763675729884159703, 2.68297506313966791050026815080, 3.58223164072139175138986386344, 4.91838388619518916977402142883, 6.19128254987258743371122786314, 6.60857182689548228374724590554, 8.696112485531838131021072172633, 9.229843429834738017743771727697, 9.790324362389914158199638704682

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