Properties

Label 2-177-1.1-c11-0-38
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  + 63.5·2-s − 243·3-s + 1.99e3·4-s + 7.69e3·5-s − 1.54e4·6-s − 8.52e4·7-s − 3.59e3·8-s + 5.90e4·9-s + 4.88e5·10-s + 5.01e5·11-s − 4.83e5·12-s + 1.19e6·13-s − 5.41e6·14-s − 1.86e6·15-s − 4.30e6·16-s − 3.66e6·17-s + 3.75e6·18-s − 1.71e6·19-s + 1.53e7·20-s + 2.07e7·21-s + 3.18e7·22-s − 8.32e5·23-s + 8.74e5·24-s + 1.03e7·25-s + 7.58e7·26-s − 1.43e7·27-s − 1.69e8·28-s + ⋯
L(s)  = 1  + 1.40·2-s − 0.577·3-s + 0.972·4-s + 1.10·5-s − 0.810·6-s − 1.91·7-s − 0.0388·8-s + 0.333·9-s + 1.54·10-s + 0.938·11-s − 0.561·12-s + 0.891·13-s − 2.69·14-s − 0.635·15-s − 1.02·16-s − 0.626·17-s + 0.468·18-s − 0.158·19-s + 1.07·20-s + 1.10·21-s + 1.31·22-s − 0.0269·23-s + 0.0224·24-s + 0.211·25-s + 1.25·26-s − 0.192·27-s − 1.86·28-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

\(L(6)\) \(\approx\) \(3.857442469\)
\(L(\frac12)\) \(\approx\) \(3.857442469\)
\(L(\frac{13}{2})\) 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 \( 1 - 63.5T + 2.04e3T^{2} \)
5 \( 1 - 7.69e3T + 4.88e7T^{2} \)
7 \( 1 + 8.52e4T + 1.97e9T^{2} \)
11 \( 1 - 5.01e5T + 2.85e11T^{2} \)
13 \( 1 - 1.19e6T + 1.79e12T^{2} \)
17 \( 1 + 3.66e6T + 3.42e13T^{2} \)
19 \( 1 + 1.71e6T + 1.16e14T^{2} \)
23 \( 1 + 8.32e5T + 9.52e14T^{2} \)
29 \( 1 - 1.09e7T + 1.22e16T^{2} \)
31 \( 1 - 2.30e8T + 2.54e16T^{2} \)
37 \( 1 - 5.65e6T + 1.77e17T^{2} \)
41 \( 1 + 2.62e8T + 5.50e17T^{2} \)
43 \( 1 - 1.38e9T + 9.29e17T^{2} \)
47 \( 1 - 1.26e9T + 2.47e18T^{2} \)
53 \( 1 + 2.59e8T + 9.26e18T^{2} \)
61 \( 1 + 3.71e9T + 4.35e19T^{2} \)
67 \( 1 - 1.36e9T + 1.22e20T^{2} \)
71 \( 1 - 1.71e10T + 2.31e20T^{2} \)
73 \( 1 - 1.87e10T + 3.13e20T^{2} \)
79 \( 1 - 1.21e10T + 7.47e20T^{2} \)
83 \( 1 - 5.60e10T + 1.28e21T^{2} \)
89 \( 1 + 4.32e9T + 2.77e21T^{2} \)
97 \( 1 + 7.04e10T + 7.15e21T^{2} \)
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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

−10.82197754476846599497274320189, −9.711202794597262566902160730635, −9.031492817702985468992851126223, −6.68609646361064146379989245574, −6.31796925640875891399189805345, −5.68578445041826053725654902240, −4.29318250716904730006201160798, −3.39647819645405043494969206743, −2.29605922925265419473190648065, −0.72658948138890686952148119275, 0.72658948138890686952148119275, 2.29605922925265419473190648065, 3.39647819645405043494969206743, 4.29318250716904730006201160798, 5.68578445041826053725654902240, 6.31796925640875891399189805345, 6.68609646361064146379989245574, 9.031492817702985468992851126223, 9.711202794597262566902160730635, 10.82197754476846599497274320189

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