Properties

Label 2-177-1.1-c11-0-54
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 70.0·2-s − 243·3-s + 2.86e3·4-s + 3.42e3·5-s + 1.70e4·6-s − 1.94e4·7-s − 5.72e4·8-s + 5.90e4·9-s − 2.40e5·10-s − 6.62e5·11-s − 6.96e5·12-s + 1.45e6·13-s + 1.36e6·14-s − 8.32e5·15-s − 1.85e6·16-s + 1.01e7·17-s − 4.13e6·18-s + 4.12e6·19-s + 9.81e6·20-s + 4.71e6·21-s + 4.64e7·22-s − 8.18e6·23-s + 1.39e7·24-s − 3.70e7·25-s − 1.01e8·26-s − 1.43e7·27-s − 5.56e7·28-s + ⋯
L(s)  = 1  − 1.54·2-s − 0.577·3-s + 1.39·4-s + 0.490·5-s + 0.894·6-s − 0.436·7-s − 0.617·8-s + 0.333·9-s − 0.759·10-s − 1.24·11-s − 0.807·12-s + 1.08·13-s + 0.676·14-s − 0.283·15-s − 0.442·16-s + 1.72·17-s − 0.516·18-s + 0.382·19-s + 0.685·20-s + 0.252·21-s + 1.92·22-s − 0.265·23-s + 0.356·24-s − 0.759·25-s − 1.68·26-s − 0.192·27-s − 0.610·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 + 70.0T + 2.04e3T^{2} \)
5 \( 1 - 3.42e3T + 4.88e7T^{2} \)
7 \( 1 + 1.94e4T + 1.97e9T^{2} \)
11 \( 1 + 6.62e5T + 2.85e11T^{2} \)
13 \( 1 - 1.45e6T + 1.79e12T^{2} \)
17 \( 1 - 1.01e7T + 3.42e13T^{2} \)
19 \( 1 - 4.12e6T + 1.16e14T^{2} \)
23 \( 1 + 8.18e6T + 9.52e14T^{2} \)
29 \( 1 + 1.12e8T + 1.22e16T^{2} \)
31 \( 1 + 6.52e7T + 2.54e16T^{2} \)
37 \( 1 + 6.08e6T + 1.77e17T^{2} \)
41 \( 1 + 6.32e8T + 5.50e17T^{2} \)
43 \( 1 - 9.00e8T + 9.29e17T^{2} \)
47 \( 1 + 1.96e9T + 2.47e18T^{2} \)
53 \( 1 - 5.33e9T + 9.26e18T^{2} \)
61 \( 1 + 6.94e9T + 4.35e19T^{2} \)
67 \( 1 + 4.15e9T + 1.22e20T^{2} \)
71 \( 1 - 1.26e10T + 2.31e20T^{2} \)
73 \( 1 - 1.33e10T + 3.13e20T^{2} \)
79 \( 1 - 3.13e10T + 7.47e20T^{2} \)
83 \( 1 - 3.96e9T + 1.28e21T^{2} \)
89 \( 1 - 3.45e10T + 2.77e21T^{2} \)
97 \( 1 - 2.67e10T + 7.15e21T^{2} \)
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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

−10.07775718870683467490692265903, −9.385925422100004290662182469009, −8.139883860208817536766749975529, −7.43895549834536392088276739581, −6.16420572572252219455860738555, −5.31953775534794197296243646193, −3.45004267569534614646321298655, −1.99231892289855929773363003208, −0.988560764940955036332289104997, 0, 0.988560764940955036332289104997, 1.99231892289855929773363003208, 3.45004267569534614646321298655, 5.31953775534794197296243646193, 6.16420572572252219455860738555, 7.43895549834536392088276739581, 8.139883860208817536766749975529, 9.385925422100004290662182469009, 10.07775718870683467490692265903

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