Properties

Label 2-177-1.1-c11-0-13
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  − 8.53·2-s − 243·3-s − 1.97e3·4-s − 453.·5-s + 2.07e3·6-s − 6.02e4·7-s + 3.43e4·8-s + 5.90e4·9-s + 3.87e3·10-s + 6.89e5·11-s + 4.79e5·12-s + 8.05e5·13-s + 5.14e5·14-s + 1.10e5·15-s + 3.75e6·16-s − 4.47e5·17-s − 5.04e5·18-s + 1.38e6·19-s + 8.95e5·20-s + 1.46e7·21-s − 5.88e6·22-s − 2.70e7·23-s − 8.34e6·24-s − 4.86e7·25-s − 6.87e6·26-s − 1.43e7·27-s + 1.19e8·28-s + ⋯
L(s)  = 1  − 0.188·2-s − 0.577·3-s − 0.964·4-s − 0.0648·5-s + 0.108·6-s − 1.35·7-s + 0.370·8-s + 0.333·9-s + 0.0122·10-s + 1.29·11-s + 0.556·12-s + 0.601·13-s + 0.255·14-s + 0.0374·15-s + 0.894·16-s − 0.0764·17-s − 0.0628·18-s + 0.128·19-s + 0.0625·20-s + 0.782·21-s − 0.243·22-s − 0.877·23-s − 0.213·24-s − 0.995·25-s − 0.113·26-s − 0.192·27-s + 1.30·28-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

\(L(6)\) \(\approx\) \(0.5265770893\)
\(L(\frac12)\) \(\approx\) \(0.5265770893\)
\(L(\frac{13}{2})\) 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 \( 1 + 8.53T + 2.04e3T^{2} \)
5 \( 1 + 453.T + 4.88e7T^{2} \)
7 \( 1 + 6.02e4T + 1.97e9T^{2} \)
11 \( 1 - 6.89e5T + 2.85e11T^{2} \)
13 \( 1 - 8.05e5T + 1.79e12T^{2} \)
17 \( 1 + 4.47e5T + 3.42e13T^{2} \)
19 \( 1 - 1.38e6T + 1.16e14T^{2} \)
23 \( 1 + 2.70e7T + 9.52e14T^{2} \)
29 \( 1 + 9.84e7T + 1.22e16T^{2} \)
31 \( 1 + 2.28e8T + 2.54e16T^{2} \)
37 \( 1 + 5.53e6T + 1.77e17T^{2} \)
41 \( 1 - 6.36e8T + 5.50e17T^{2} \)
43 \( 1 + 1.32e9T + 9.29e17T^{2} \)
47 \( 1 - 2.16e9T + 2.47e18T^{2} \)
53 \( 1 - 1.51e8T + 9.26e18T^{2} \)
61 \( 1 + 1.17e10T + 4.35e19T^{2} \)
67 \( 1 + 1.16e10T + 1.22e20T^{2} \)
71 \( 1 - 2.77e10T + 2.31e20T^{2} \)
73 \( 1 + 9.19e9T + 3.13e20T^{2} \)
79 \( 1 + 2.72e10T + 7.47e20T^{2} \)
83 \( 1 - 1.20e10T + 1.28e21T^{2} \)
89 \( 1 - 5.23e10T + 2.77e21T^{2} \)
97 \( 1 - 1.76e10T + 7.15e21T^{2} \)
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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

−10.49427054981386082104225917340, −9.508104264002617471066185109323, −9.028914504491465515483024627906, −7.59482438536735917914086185415, −6.38585648565935682637473856625, −5.64088201348462964206386590151, −4.11535503566280356497571730194, −3.54575838560009690443015285647, −1.57280911741665368818511158239, −0.36426243508823276666083051738, 0.36426243508823276666083051738, 1.57280911741665368818511158239, 3.54575838560009690443015285647, 4.11535503566280356497571730194, 5.64088201348462964206386590151, 6.38585648565935682637473856625, 7.59482438536735917914086185415, 9.028914504491465515483024627906, 9.508104264002617471066185109323, 10.49427054981386082104225917340

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