Properties

Label 2-177-1.1-c11-0-21
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 31.7·2-s + 243·3-s − 1.04e3·4-s + 1.82e3·5-s + 7.71e3·6-s − 5.20e4·7-s − 9.80e4·8-s + 5.90e4·9-s + 5.78e4·10-s − 4.43e5·11-s − 2.53e5·12-s + 7.80e5·13-s − 1.65e6·14-s + 4.43e5·15-s − 9.77e5·16-s + 6.30e6·17-s + 1.87e6·18-s − 9.72e6·19-s − 1.89e6·20-s − 1.26e7·21-s − 1.40e7·22-s − 1.54e7·23-s − 2.38e7·24-s − 4.55e7·25-s + 2.47e7·26-s + 1.43e7·27-s + 5.41e7·28-s + ⋯
L(s)  = 1  + 0.701·2-s + 0.577·3-s − 0.508·4-s + 0.260·5-s + 0.404·6-s − 1.17·7-s − 1.05·8-s + 0.333·9-s + 0.182·10-s − 0.829·11-s − 0.293·12-s + 0.582·13-s − 0.820·14-s + 0.150·15-s − 0.233·16-s + 1.07·17-s + 0.233·18-s − 0.901·19-s − 0.132·20-s − 0.675·21-s − 0.581·22-s − 0.501·23-s − 0.610·24-s − 0.931·25-s + 0.408·26-s + 0.192·27-s + 0.594·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(2.144858591\)
\(L(\frac12)\) \(\approx\) \(2.144858591\)
\(L(\frac{13}{2})\) 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 \( 1 - 31.7T + 2.04e3T^{2} \)
5 \( 1 - 1.82e3T + 4.88e7T^{2} \)
7 \( 1 + 5.20e4T + 1.97e9T^{2} \)
11 \( 1 + 4.43e5T + 2.85e11T^{2} \)
13 \( 1 - 7.80e5T + 1.79e12T^{2} \)
17 \( 1 - 6.30e6T + 3.42e13T^{2} \)
19 \( 1 + 9.72e6T + 1.16e14T^{2} \)
23 \( 1 + 1.54e7T + 9.52e14T^{2} \)
29 \( 1 + 1.80e8T + 1.22e16T^{2} \)
31 \( 1 - 2.28e8T + 2.54e16T^{2} \)
37 \( 1 + 3.90e8T + 1.77e17T^{2} \)
41 \( 1 - 7.69e8T + 5.50e17T^{2} \)
43 \( 1 - 2.12e8T + 9.29e17T^{2} \)
47 \( 1 - 1.63e9T + 2.47e18T^{2} \)
53 \( 1 - 4.66e9T + 9.26e18T^{2} \)
61 \( 1 - 1.49e9T + 4.35e19T^{2} \)
67 \( 1 - 1.66e8T + 1.22e20T^{2} \)
71 \( 1 + 1.80e10T + 2.31e20T^{2} \)
73 \( 1 - 1.81e10T + 3.13e20T^{2} \)
79 \( 1 - 1.92e10T + 7.47e20T^{2} \)
83 \( 1 - 3.05e10T + 1.28e21T^{2} \)
89 \( 1 + 8.33e9T + 2.77e21T^{2} \)
97 \( 1 - 8.42e10T + 7.15e21T^{2} \)
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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

−10.40232765221867296853885924156, −9.662004393725168317261383158836, −8.755335614499393910383050964920, −7.68193382435137232104360847758, −6.23793627530253523043571578264, −5.50944661663125830699610633306, −4.07855579174573129123671227138, −3.35083253129199590386269972427, −2.29127842565930026613369190244, −0.55859793436134671317424367390, 0.55859793436134671317424367390, 2.29127842565930026613369190244, 3.35083253129199590386269972427, 4.07855579174573129123671227138, 5.50944661663125830699610633306, 6.23793627530253523043571578264, 7.68193382435137232104360847758, 8.755335614499393910383050964920, 9.662004393725168317261383158836, 10.40232765221867296853885924156

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