Properties

Label 2-177-1.1-c9-0-11
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  + 25.6·2-s − 81·3-s + 147.·4-s − 2.01e3·5-s − 2.07e3·6-s + 7.28e3·7-s − 9.36e3·8-s + 6.56e3·9-s − 5.18e4·10-s − 8.61e4·11-s − 1.19e4·12-s + 1.21e5·13-s + 1.86e5·14-s + 1.63e5·15-s − 3.15e5·16-s − 2.73e5·17-s + 1.68e5·18-s − 1.10e6·19-s − 2.96e5·20-s − 5.89e5·21-s − 2.21e6·22-s − 1.28e5·23-s + 7.58e5·24-s + 2.12e6·25-s + 3.11e6·26-s − 5.31e5·27-s + 1.07e6·28-s + ⋯
L(s)  = 1  + 1.13·2-s − 0.577·3-s + 0.287·4-s − 1.44·5-s − 0.655·6-s + 1.14·7-s − 0.808·8-s + 0.333·9-s − 1.63·10-s − 1.77·11-s − 0.165·12-s + 1.17·13-s + 1.30·14-s + 0.834·15-s − 1.20·16-s − 0.794·17-s + 0.378·18-s − 1.95·19-s − 0.414·20-s − 0.661·21-s − 2.01·22-s − 0.0959·23-s + 0.466·24-s + 1.08·25-s + 1.33·26-s − 0.192·27-s + 0.329·28-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

\(L(5)\) \(\approx\) \(1.235412907\)
\(L(\frac12)\) \(\approx\) \(1.235412907\)
\(L(\frac{11}{2})\) 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 \( 1 - 25.6T + 512T^{2} \)
5 \( 1 + 2.01e3T + 1.95e6T^{2} \)
7 \( 1 - 7.28e3T + 4.03e7T^{2} \)
11 \( 1 + 8.61e4T + 2.35e9T^{2} \)
13 \( 1 - 1.21e5T + 1.06e10T^{2} \)
17 \( 1 + 2.73e5T + 1.18e11T^{2} \)
19 \( 1 + 1.10e6T + 3.22e11T^{2} \)
23 \( 1 + 1.28e5T + 1.80e12T^{2} \)
29 \( 1 - 9.54e5T + 1.45e13T^{2} \)
31 \( 1 - 1.83e6T + 2.64e13T^{2} \)
37 \( 1 - 5.14e6T + 1.29e14T^{2} \)
41 \( 1 + 1.62e7T + 3.27e14T^{2} \)
43 \( 1 - 4.98e4T + 5.02e14T^{2} \)
47 \( 1 - 5.78e7T + 1.11e15T^{2} \)
53 \( 1 - 1.08e7T + 3.29e15T^{2} \)
61 \( 1 - 1.48e8T + 1.16e16T^{2} \)
67 \( 1 + 2.17e7T + 2.72e16T^{2} \)
71 \( 1 - 1.65e8T + 4.58e16T^{2} \)
73 \( 1 - 2.22e8T + 5.88e16T^{2} \)
79 \( 1 - 4.50e8T + 1.19e17T^{2} \)
83 \( 1 - 1.85e8T + 1.86e17T^{2} \)
89 \( 1 + 6.59e8T + 3.50e17T^{2} \)
97 \( 1 - 1.23e9T + 7.60e17T^{2} \)
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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

−11.11028425694599517894037362071, −10.72736852892139277108827725893, −8.542091407260340628302932330138, −8.033531822503587550864047628244, −6.62577127530846732950309416833, −5.39509572407964342936765017114, −4.52723972173394254177431991750, −3.88725093654844894504738996081, −2.37235089602674652593094554241, −0.45604288739540236439456326909, 0.45604288739540236439456326909, 2.37235089602674652593094554241, 3.88725093654844894504738996081, 4.52723972173394254177431991750, 5.39509572407964342936765017114, 6.62577127530846732950309416833, 8.033531822503587550864047628244, 8.542091407260340628302932330138, 10.72736852892139277108827725893, 11.11028425694599517894037362071

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