Properties

Label 2-177-1.1-c11-0-15
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  − 58.2·2-s − 243·3-s + 1.34e3·4-s + 1.17e4·5-s + 1.41e4·6-s + 2.45e4·7-s + 4.10e4·8-s + 5.90e4·9-s − 6.85e5·10-s − 9.38e5·11-s − 3.26e5·12-s − 1.32e6·13-s − 1.43e6·14-s − 2.85e6·15-s − 5.14e6·16-s + 5.00e5·17-s − 3.43e6·18-s − 1.82e7·19-s + 1.57e7·20-s − 5.97e6·21-s + 5.46e7·22-s − 3.11e6·23-s − 9.98e6·24-s + 8.95e7·25-s + 7.74e7·26-s − 1.43e7·27-s + 3.30e7·28-s + ⋯
L(s)  = 1  − 1.28·2-s − 0.577·3-s + 0.655·4-s + 1.68·5-s + 0.742·6-s + 0.552·7-s + 0.443·8-s + 0.333·9-s − 2.16·10-s − 1.75·11-s − 0.378·12-s − 0.992·13-s − 0.711·14-s − 0.972·15-s − 1.22·16-s + 0.0855·17-s − 0.428·18-s − 1.69·19-s + 1.10·20-s − 0.319·21-s + 2.26·22-s − 0.100·23-s − 0.255·24-s + 1.83·25-s + 1.27·26-s − 0.192·27-s + 0.362·28-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

\(L(6)\) \(\approx\) \(0.6785171520\)
\(L(\frac12)\) \(\approx\) \(0.6785171520\)
\(L(\frac{13}{2})\) 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 \( 1 + 58.2T + 2.04e3T^{2} \)
5 \( 1 - 1.17e4T + 4.88e7T^{2} \)
7 \( 1 - 2.45e4T + 1.97e9T^{2} \)
11 \( 1 + 9.38e5T + 2.85e11T^{2} \)
13 \( 1 + 1.32e6T + 1.79e12T^{2} \)
17 \( 1 - 5.00e5T + 3.42e13T^{2} \)
19 \( 1 + 1.82e7T + 1.16e14T^{2} \)
23 \( 1 + 3.11e6T + 9.52e14T^{2} \)
29 \( 1 + 1.34e8T + 1.22e16T^{2} \)
31 \( 1 - 1.46e8T + 2.54e16T^{2} \)
37 \( 1 + 7.84e8T + 1.77e17T^{2} \)
41 \( 1 - 8.56e8T + 5.50e17T^{2} \)
43 \( 1 + 1.12e9T + 9.29e17T^{2} \)
47 \( 1 - 1.67e9T + 2.47e18T^{2} \)
53 \( 1 + 1.57e9T + 9.26e18T^{2} \)
61 \( 1 - 1.01e10T + 4.35e19T^{2} \)
67 \( 1 - 1.64e10T + 1.22e20T^{2} \)
71 \( 1 - 1.39e10T + 2.31e20T^{2} \)
73 \( 1 + 2.39e10T + 3.13e20T^{2} \)
79 \( 1 + 4.32e9T + 7.47e20T^{2} \)
83 \( 1 - 4.65e10T + 1.28e21T^{2} \)
89 \( 1 + 3.36e10T + 2.77e21T^{2} \)
97 \( 1 - 1.01e11T + 7.15e21T^{2} \)
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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

−10.32274122604648584858033612961, −9.911348710452175737182858685744, −8.785279299149834227674840857203, −7.81224690337722131491483978522, −6.71532046148091186745599482911, −5.48755895445109827160918241248, −4.78674360332369612827780535733, −2.32136169588672391161764108844, −1.85342878344987950976812155268, −0.45336343576841676752129191905, 0.45336343576841676752129191905, 1.85342878344987950976812155268, 2.32136169588672391161764108844, 4.78674360332369612827780535733, 5.48755895445109827160918241248, 6.71532046148091186745599482911, 7.81224690337722131491483978522, 8.785279299149834227674840857203, 9.911348710452175737182858685744, 10.32274122604648584858033612961

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