Properties

Label 2-177-1.1-c9-0-47
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.4·2-s + 81·3-s − 272.·4-s + 2.70e3·5-s + 1.25e3·6-s + 6.16e3·7-s − 1.21e4·8-s + 6.56e3·9-s + 4.18e4·10-s − 3.28e4·11-s − 2.20e4·12-s + 5.11e4·13-s + 9.54e4·14-s + 2.19e5·15-s − 4.85e4·16-s + 3.15e5·17-s + 1.01e5·18-s − 3.44e5·19-s − 7.36e5·20-s + 4.99e5·21-s − 5.08e5·22-s + 1.91e6·23-s − 9.83e5·24-s + 5.36e6·25-s + 7.91e5·26-s + 5.31e5·27-s − 1.67e6·28-s + ⋯
L(s)  = 1  + 0.684·2-s + 0.577·3-s − 0.531·4-s + 1.93·5-s + 0.394·6-s + 0.970·7-s − 1.04·8-s + 0.333·9-s + 1.32·10-s − 0.676·11-s − 0.307·12-s + 0.496·13-s + 0.664·14-s + 1.11·15-s − 0.185·16-s + 0.917·17-s + 0.228·18-s − 0.606·19-s − 1.02·20-s + 0.560·21-s − 0.463·22-s + 1.42·23-s − 0.605·24-s + 2.74·25-s + 0.339·26-s + 0.192·27-s − 0.516·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(5.603182771\)
\(L(\frac12)\) \(\approx\) \(5.603182771\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 - 15.4T + 512T^{2} \)
5 \( 1 - 2.70e3T + 1.95e6T^{2} \)
7 \( 1 - 6.16e3T + 4.03e7T^{2} \)
11 \( 1 + 3.28e4T + 2.35e9T^{2} \)
13 \( 1 - 5.11e4T + 1.06e10T^{2} \)
17 \( 1 - 3.15e5T + 1.18e11T^{2} \)
19 \( 1 + 3.44e5T + 3.22e11T^{2} \)
23 \( 1 - 1.91e6T + 1.80e12T^{2} \)
29 \( 1 - 9.38e4T + 1.45e13T^{2} \)
31 \( 1 + 1.74e6T + 2.64e13T^{2} \)
37 \( 1 - 5.42e6T + 1.29e14T^{2} \)
41 \( 1 + 2.76e7T + 3.27e14T^{2} \)
43 \( 1 + 3.64e7T + 5.02e14T^{2} \)
47 \( 1 - 3.13e7T + 1.11e15T^{2} \)
53 \( 1 - 2.10e7T + 3.29e15T^{2} \)
61 \( 1 + 2.14e7T + 1.16e16T^{2} \)
67 \( 1 - 1.80e8T + 2.72e16T^{2} \)
71 \( 1 + 1.63e8T + 4.58e16T^{2} \)
73 \( 1 - 3.32e8T + 5.88e16T^{2} \)
79 \( 1 - 5.93e8T + 1.19e17T^{2} \)
83 \( 1 - 6.43e8T + 1.86e17T^{2} \)
89 \( 1 + 5.28e7T + 3.50e17T^{2} \)
97 \( 1 + 6.12e8T + 7.60e17T^{2} \)
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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

−10.83959823313234623571122847214, −9.918292284048216111788489845659, −9.035781914581144122914868277486, −8.214905310238428714815304528484, −6.57648005433134337275776916556, −5.40385807092824150645515504771, −4.91902339879200635043194216303, −3.29869982147003716217445803584, −2.19158322991846397667022540683, −1.12791342597850130003805843750, 1.12791342597850130003805843750, 2.19158322991846397667022540683, 3.29869982147003716217445803584, 4.91902339879200635043194216303, 5.40385807092824150645515504771, 6.57648005433134337275776916556, 8.214905310238428714815304528484, 9.035781914581144122914868277486, 9.918292284048216111788489845659, 10.83959823313234623571122847214

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