Properties

Label 2-177-1.1-c11-0-50
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  − 47.6·2-s + 243·3-s + 226.·4-s + 1.00e4·5-s − 1.15e4·6-s + 1.98e4·7-s + 8.68e4·8-s + 5.90e4·9-s − 4.80e5·10-s − 2.56e5·11-s + 5.50e4·12-s + 1.87e6·13-s − 9.48e5·14-s + 2.44e6·15-s − 4.60e6·16-s + 3.91e6·17-s − 2.81e6·18-s − 1.36e7·19-s + 2.28e6·20-s + 4.83e6·21-s + 1.22e7·22-s + 4.09e7·23-s + 2.11e7·24-s + 5.25e7·25-s − 8.93e7·26-s + 1.43e7·27-s + 4.50e6·28-s + ⋯
L(s)  = 1  − 1.05·2-s + 0.577·3-s + 0.110·4-s + 1.44·5-s − 0.608·6-s + 0.447·7-s + 0.937·8-s + 0.333·9-s − 1.51·10-s − 0.480·11-s + 0.0638·12-s + 1.39·13-s − 0.471·14-s + 0.831·15-s − 1.09·16-s + 0.669·17-s − 0.351·18-s − 1.26·19-s + 0.159·20-s + 0.258·21-s + 0.506·22-s + 1.32·23-s + 0.541·24-s + 1.07·25-s − 1.47·26-s + 0.192·27-s + 0.0495·28-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

\(L(6)\) \(\approx\) \(2.544593403\)
\(L(\frac12)\) \(\approx\) \(2.544593403\)
\(L(\frac{13}{2})\) 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 \( 1 + 47.6T + 2.04e3T^{2} \)
5 \( 1 - 1.00e4T + 4.88e7T^{2} \)
7 \( 1 - 1.98e4T + 1.97e9T^{2} \)
11 \( 1 + 2.56e5T + 2.85e11T^{2} \)
13 \( 1 - 1.87e6T + 1.79e12T^{2} \)
17 \( 1 - 3.91e6T + 3.42e13T^{2} \)
19 \( 1 + 1.36e7T + 1.16e14T^{2} \)
23 \( 1 - 4.09e7T + 9.52e14T^{2} \)
29 \( 1 + 2.21e7T + 1.22e16T^{2} \)
31 \( 1 - 2.03e8T + 2.54e16T^{2} \)
37 \( 1 - 7.60e8T + 1.77e17T^{2} \)
41 \( 1 + 1.05e9T + 5.50e17T^{2} \)
43 \( 1 - 6.64e8T + 9.29e17T^{2} \)
47 \( 1 - 1.48e9T + 2.47e18T^{2} \)
53 \( 1 - 9.91e8T + 9.26e18T^{2} \)
61 \( 1 - 8.36e9T + 4.35e19T^{2} \)
67 \( 1 + 1.21e10T + 1.22e20T^{2} \)
71 \( 1 - 4.45e9T + 2.31e20T^{2} \)
73 \( 1 - 6.92e9T + 3.13e20T^{2} \)
79 \( 1 + 1.11e9T + 7.47e20T^{2} \)
83 \( 1 + 6.93e10T + 1.28e21T^{2} \)
89 \( 1 + 4.21e10T + 2.77e21T^{2} \)
97 \( 1 + 7.55e10T + 7.15e21T^{2} \)
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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

−10.35033012154585396040535501861, −9.578923813242669461544873048900, −8.689376924388319153787947411850, −8.099628110131616210789228766727, −6.74163197140721537593040483979, −5.59011127697149809351508507850, −4.33729216012998174365125634592, −2.70419938328273308762959804635, −1.63707890496646135392091449030, −0.911609703513889432939882230665, 0.911609703513889432939882230665, 1.63707890496646135392091449030, 2.70419938328273308762959804635, 4.33729216012998174365125634592, 5.59011127697149809351508507850, 6.74163197140721537593040483979, 8.099628110131616210789228766727, 8.689376924388319153787947411850, 9.578923813242669461544873048900, 10.35033012154585396040535501861

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