Properties

Label 2-177-1.1-c11-0-64
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  + 10.5·2-s + 243·3-s − 1.93e3·4-s + 8.30e3·5-s + 2.55e3·6-s + 4.64e4·7-s − 4.19e4·8-s + 5.90e4·9-s + 8.74e4·10-s + 5.29e5·11-s − 4.70e5·12-s + 1.93e6·13-s + 4.89e5·14-s + 2.01e6·15-s + 3.52e6·16-s + 5.64e6·17-s + 6.21e5·18-s + 2.69e6·19-s − 1.60e7·20-s + 1.12e7·21-s + 5.57e6·22-s − 1.76e7·23-s − 1.01e7·24-s + 2.00e7·25-s + 2.03e7·26-s + 1.43e7·27-s − 9.00e7·28-s + ⋯
L(s)  = 1  + 0.232·2-s + 0.577·3-s − 0.945·4-s + 1.18·5-s + 0.134·6-s + 1.04·7-s − 0.452·8-s + 0.333·9-s + 0.276·10-s + 0.991·11-s − 0.546·12-s + 1.44·13-s + 0.243·14-s + 0.685·15-s + 0.840·16-s + 0.964·17-s + 0.0775·18-s + 0.250·19-s − 1.12·20-s + 0.603·21-s + 0.230·22-s − 0.570·23-s − 0.261·24-s + 0.410·25-s + 0.335·26-s + 0.192·27-s − 0.988·28-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

\(L(6)\) \(\approx\) \(4.761624096\)
\(L(\frac12)\) \(\approx\) \(4.761624096\)
\(L(\frac{13}{2})\) 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 \( 1 - 10.5T + 2.04e3T^{2} \)
5 \( 1 - 8.30e3T + 4.88e7T^{2} \)
7 \( 1 - 4.64e4T + 1.97e9T^{2} \)
11 \( 1 - 5.29e5T + 2.85e11T^{2} \)
13 \( 1 - 1.93e6T + 1.79e12T^{2} \)
17 \( 1 - 5.64e6T + 3.42e13T^{2} \)
19 \( 1 - 2.69e6T + 1.16e14T^{2} \)
23 \( 1 + 1.76e7T + 9.52e14T^{2} \)
29 \( 1 - 1.24e8T + 1.22e16T^{2} \)
31 \( 1 + 1.09e8T + 2.54e16T^{2} \)
37 \( 1 + 1.90e8T + 1.77e17T^{2} \)
41 \( 1 - 2.79e8T + 5.50e17T^{2} \)
43 \( 1 - 5.67e8T + 9.29e17T^{2} \)
47 \( 1 + 1.78e9T + 2.47e18T^{2} \)
53 \( 1 + 5.80e8T + 9.26e18T^{2} \)
61 \( 1 + 2.90e9T + 4.35e19T^{2} \)
67 \( 1 + 5.69e9T + 1.22e20T^{2} \)
71 \( 1 + 4.85e9T + 2.31e20T^{2} \)
73 \( 1 + 5.37e9T + 3.13e20T^{2} \)
79 \( 1 - 1.77e10T + 7.47e20T^{2} \)
83 \( 1 + 1.09e10T + 1.28e21T^{2} \)
89 \( 1 + 1.49e10T + 2.77e21T^{2} \)
97 \( 1 - 7.14e10T + 7.15e21T^{2} \)
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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

−10.43691629441439843874862901391, −9.501049527373166193984842822928, −8.757980764175067944091451518127, −7.939514966048712045212145184987, −6.29316035180122904868263869920, −5.42670393599044005555783374303, −4.28961518463977270404316366696, −3.29514874234404554600922721195, −1.70264210979893655968410467940, −1.07801099843553702569277593755, 1.07801099843553702569277593755, 1.70264210979893655968410467940, 3.29514874234404554600922721195, 4.28961518463977270404316366696, 5.42670393599044005555783374303, 6.29316035180122904868263869920, 7.939514966048712045212145184987, 8.757980764175067944091451518127, 9.501049527373166193984842822928, 10.43691629441439843874862901391

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