Properties

Label 2-177-1.1-c11-0-30
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  − 88.1·2-s + 243·3-s + 5.71e3·4-s + 1.09e3·5-s − 2.14e4·6-s − 2.05e4·7-s − 3.23e5·8-s + 5.90e4·9-s − 9.63e4·10-s + 6.19e5·11-s + 1.38e6·12-s + 1.84e6·13-s + 1.80e6·14-s + 2.65e5·15-s + 1.67e7·16-s − 6.92e6·17-s − 5.20e6·18-s − 1.82e7·19-s + 6.24e6·20-s − 4.98e6·21-s − 5.45e7·22-s + 3.56e7·23-s − 7.84e7·24-s − 4.76e7·25-s − 1.62e8·26-s + 1.43e7·27-s − 1.17e8·28-s + ⋯
L(s)  = 1  − 1.94·2-s + 0.577·3-s + 2.79·4-s + 0.156·5-s − 1.12·6-s − 0.461·7-s − 3.48·8-s + 0.333·9-s − 0.304·10-s + 1.15·11-s + 1.61·12-s + 1.38·13-s + 0.898·14-s + 0.0903·15-s + 3.99·16-s − 1.18·17-s − 0.648·18-s − 1.68·19-s + 0.436·20-s − 0.266·21-s − 2.25·22-s + 1.15·23-s − 2.01·24-s − 0.975·25-s − 2.68·26-s + 0.192·27-s − 1.28·28-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

\(L(6)\) \(\approx\) \(1.187363702\)
\(L(\frac12)\) \(\approx\) \(1.187363702\)
\(L(\frac{13}{2})\) 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 \( 1 + 88.1T + 2.04e3T^{2} \)
5 \( 1 - 1.09e3T + 4.88e7T^{2} \)
7 \( 1 + 2.05e4T + 1.97e9T^{2} \)
11 \( 1 - 6.19e5T + 2.85e11T^{2} \)
13 \( 1 - 1.84e6T + 1.79e12T^{2} \)
17 \( 1 + 6.92e6T + 3.42e13T^{2} \)
19 \( 1 + 1.82e7T + 1.16e14T^{2} \)
23 \( 1 - 3.56e7T + 9.52e14T^{2} \)
29 \( 1 - 1.95e8T + 1.22e16T^{2} \)
31 \( 1 + 2.04e7T + 2.54e16T^{2} \)
37 \( 1 + 1.33e8T + 1.77e17T^{2} \)
41 \( 1 - 5.40e8T + 5.50e17T^{2} \)
43 \( 1 - 6.33e8T + 9.29e17T^{2} \)
47 \( 1 - 1.05e8T + 2.47e18T^{2} \)
53 \( 1 - 2.39e9T + 9.26e18T^{2} \)
61 \( 1 + 5.42e9T + 4.35e19T^{2} \)
67 \( 1 - 3.68e9T + 1.22e20T^{2} \)
71 \( 1 + 1.73e10T + 2.31e20T^{2} \)
73 \( 1 - 3.22e10T + 3.13e20T^{2} \)
79 \( 1 - 3.35e10T + 7.47e20T^{2} \)
83 \( 1 - 4.98e10T + 1.28e21T^{2} \)
89 \( 1 - 4.19e10T + 2.77e21T^{2} \)
97 \( 1 - 5.06e10T + 7.15e21T^{2} \)
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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

−10.50302536700949396308341494929, −9.258397489652167994255673555838, −8.886401692899856694038093302243, −8.064942636895330527467701700066, −6.58385183971897654128256085119, −6.43053600541791662663695064385, −3.88478769766252622009512182472, −2.59762459695695822630369712498, −1.62073837096125984285310072541, −0.66331899974088398369298069311, 0.66331899974088398369298069311, 1.62073837096125984285310072541, 2.59762459695695822630369712498, 3.88478769766252622009512182472, 6.43053600541791662663695064385, 6.58385183971897654128256085119, 8.064942636895330527467701700066, 8.886401692899856694038093302243, 9.258397489652167994255673555838, 10.50302536700949396308341494929

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