Properties

Label 2-177-1.1-c11-0-66
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  + 52.4·2-s + 243·3-s + 700.·4-s + 6.87e3·5-s + 1.27e4·6-s + 3.59e4·7-s − 7.06e4·8-s + 5.90e4·9-s + 3.60e5·10-s + 2.39e5·11-s + 1.70e5·12-s + 1.50e6·13-s + 1.88e6·14-s + 1.66e6·15-s − 5.13e6·16-s − 6.10e6·17-s + 3.09e6·18-s + 1.76e7·19-s + 4.81e6·20-s + 8.73e6·21-s + 1.25e7·22-s + 3.61e7·23-s − 1.71e7·24-s − 1.60e6·25-s + 7.88e7·26-s + 1.43e7·27-s + 2.51e7·28-s + ⋯
L(s)  = 1  + 1.15·2-s + 0.577·3-s + 0.341·4-s + 0.983·5-s + 0.668·6-s + 0.808·7-s − 0.762·8-s + 0.333·9-s + 1.13·10-s + 0.447·11-s + 0.197·12-s + 1.12·13-s + 0.936·14-s + 0.567·15-s − 1.22·16-s − 1.04·17-s + 0.386·18-s + 1.63·19-s + 0.336·20-s + 0.466·21-s + 0.518·22-s + 1.17·23-s − 0.440·24-s − 0.0328·25-s + 1.30·26-s + 0.192·27-s + 0.276·28-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

\(L(6)\) \(\approx\) \(7.418510051\)
\(L(\frac12)\) \(\approx\) \(7.418510051\)
\(L(\frac{13}{2})\) 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 \( 1 - 52.4T + 2.04e3T^{2} \)
5 \( 1 - 6.87e3T + 4.88e7T^{2} \)
7 \( 1 - 3.59e4T + 1.97e9T^{2} \)
11 \( 1 - 2.39e5T + 2.85e11T^{2} \)
13 \( 1 - 1.50e6T + 1.79e12T^{2} \)
17 \( 1 + 6.10e6T + 3.42e13T^{2} \)
19 \( 1 - 1.76e7T + 1.16e14T^{2} \)
23 \( 1 - 3.61e7T + 9.52e14T^{2} \)
29 \( 1 + 1.35e8T + 1.22e16T^{2} \)
31 \( 1 + 1.11e8T + 2.54e16T^{2} \)
37 \( 1 - 7.49e8T + 1.77e17T^{2} \)
41 \( 1 + 2.08e8T + 5.50e17T^{2} \)
43 \( 1 - 3.45e8T + 9.29e17T^{2} \)
47 \( 1 - 2.38e9T + 2.47e18T^{2} \)
53 \( 1 - 2.51e9T + 9.26e18T^{2} \)
61 \( 1 + 1.11e10T + 4.35e19T^{2} \)
67 \( 1 + 4.24e9T + 1.22e20T^{2} \)
71 \( 1 + 4.21e9T + 2.31e20T^{2} \)
73 \( 1 - 1.27e10T + 3.13e20T^{2} \)
79 \( 1 + 3.34e10T + 7.47e20T^{2} \)
83 \( 1 - 2.64e10T + 1.28e21T^{2} \)
89 \( 1 - 6.04e10T + 2.77e21T^{2} \)
97 \( 1 - 1.29e11T + 7.15e21T^{2} \)
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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

−10.93311519651197650447151735694, −9.357852882499834253473997430598, −8.942707829816485098041497241753, −7.47381658940560527272041056150, −6.18346147270908680507199107453, −5.37406684290906411176423815910, −4.32092416794711810250500203301, −3.30764841280480524995277445383, −2.16260500984583392076723254886, −1.09669647993396878110403401943, 1.09669647993396878110403401943, 2.16260500984583392076723254886, 3.30764841280480524995277445383, 4.32092416794711810250500203301, 5.37406684290906411176423815910, 6.18346147270908680507199107453, 7.47381658940560527272041056150, 8.942707829816485098041497241753, 9.357852882499834253473997430598, 10.93311519651197650447151735694

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