Properties

Label 2-177-1.1-c9-0-20
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  + 17.2·2-s − 81·3-s − 215.·4-s + 27.4·5-s − 1.39e3·6-s + 8.66e3·7-s − 1.25e4·8-s + 6.56e3·9-s + 473.·10-s + 4.23e4·11-s + 1.74e4·12-s − 2.00e5·13-s + 1.49e5·14-s − 2.22e3·15-s − 1.05e5·16-s + 1.54e5·17-s + 1.13e5·18-s − 7.24e4·19-s − 5.91e3·20-s − 7.01e5·21-s + 7.29e5·22-s + 1.79e6·23-s + 1.01e6·24-s − 1.95e6·25-s − 3.45e6·26-s − 5.31e5·27-s − 1.86e6·28-s + ⋯
L(s)  = 1  + 0.761·2-s − 0.577·3-s − 0.420·4-s + 0.0196·5-s − 0.439·6-s + 1.36·7-s − 1.08·8-s + 0.333·9-s + 0.0149·10-s + 0.871·11-s + 0.242·12-s − 1.94·13-s + 1.03·14-s − 0.0113·15-s − 0.402·16-s + 0.449·17-s + 0.253·18-s − 0.127·19-s − 0.00826·20-s − 0.787·21-s + 0.663·22-s + 1.33·23-s + 0.624·24-s − 0.999·25-s − 1.48·26-s − 0.192·27-s − 0.573·28-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

\(L(5)\) \(\approx\) \(2.222499078\)
\(L(\frac12)\) \(\approx\) \(2.222499078\)
\(L(\frac{11}{2})\) 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 \( 1 - 17.2T + 512T^{2} \)
5 \( 1 - 27.4T + 1.95e6T^{2} \)
7 \( 1 - 8.66e3T + 4.03e7T^{2} \)
11 \( 1 - 4.23e4T + 2.35e9T^{2} \)
13 \( 1 + 2.00e5T + 1.06e10T^{2} \)
17 \( 1 - 1.54e5T + 1.18e11T^{2} \)
19 \( 1 + 7.24e4T + 3.22e11T^{2} \)
23 \( 1 - 1.79e6T + 1.80e12T^{2} \)
29 \( 1 - 2.62e6T + 1.45e13T^{2} \)
31 \( 1 + 6.56e6T + 2.64e13T^{2} \)
37 \( 1 - 2.16e6T + 1.29e14T^{2} \)
41 \( 1 + 1.12e7T + 3.27e14T^{2} \)
43 \( 1 + 3.59e7T + 5.02e14T^{2} \)
47 \( 1 - 2.37e7T + 1.11e15T^{2} \)
53 \( 1 - 8.03e6T + 3.29e15T^{2} \)
61 \( 1 - 1.76e8T + 1.16e16T^{2} \)
67 \( 1 - 1.31e8T + 2.72e16T^{2} \)
71 \( 1 - 3.10e8T + 4.58e16T^{2} \)
73 \( 1 - 2.50e8T + 5.88e16T^{2} \)
79 \( 1 - 3.50e8T + 1.19e17T^{2} \)
83 \( 1 - 3.42e8T + 1.86e17T^{2} \)
89 \( 1 - 8.59e8T + 3.50e17T^{2} \)
97 \( 1 - 1.29e9T + 7.60e17T^{2} \)
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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

−11.41536835699110499740331992844, −10.04908385141159413991123110427, −9.081535055375027827028389880327, −7.84279077712803429121746822809, −6.71115431284799774125780601022, −5.20289036395324943039239211416, −4.93582449853109097561233876825, −3.72378252115489084426848292129, −2.09526810219923742498295389477, −0.68075729435273914878525737609, 0.68075729435273914878525737609, 2.09526810219923742498295389477, 3.72378252115489084426848292129, 4.93582449853109097561233876825, 5.20289036395324943039239211416, 6.71115431284799774125780601022, 7.84279077712803429121746822809, 9.081535055375027827028389880327, 10.04908385141159413991123110427, 11.41536835699110499740331992844

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