Properties

Label 2-177-1.1-c9-0-30
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  − 20.5·2-s − 81·3-s − 90.7·4-s − 622.·5-s + 1.66e3·6-s + 1.39e3·7-s + 1.23e4·8-s + 6.56e3·9-s + 1.27e4·10-s + 8.19e4·11-s + 7.35e3·12-s + 2.21e4·13-s − 2.86e4·14-s + 5.04e4·15-s − 2.07e5·16-s + 5.12e5·17-s − 1.34e5·18-s + 8.50e5·19-s + 5.65e4·20-s − 1.12e5·21-s − 1.68e6·22-s + 1.92e6·23-s − 1.00e6·24-s − 1.56e6·25-s − 4.54e5·26-s − 5.31e5·27-s − 1.26e5·28-s + ⋯
L(s)  = 1  − 0.907·2-s − 0.577·3-s − 0.177·4-s − 0.445·5-s + 0.523·6-s + 0.219·7-s + 1.06·8-s + 0.333·9-s + 0.404·10-s + 1.68·11-s + 0.102·12-s + 0.215·13-s − 0.199·14-s + 0.257·15-s − 0.791·16-s + 1.48·17-s − 0.302·18-s + 1.49·19-s + 0.0790·20-s − 0.126·21-s − 1.53·22-s + 1.43·23-s − 0.616·24-s − 0.801·25-s − 0.195·26-s − 0.192·27-s − 0.0389·28-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

\(L(5)\) \(\approx\) \(1.280626872\)
\(L(\frac12)\) \(\approx\) \(1.280626872\)
\(L(\frac{11}{2})\) 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 \( 1 + 20.5T + 512T^{2} \)
5 \( 1 + 622.T + 1.95e6T^{2} \)
7 \( 1 - 1.39e3T + 4.03e7T^{2} \)
11 \( 1 - 8.19e4T + 2.35e9T^{2} \)
13 \( 1 - 2.21e4T + 1.06e10T^{2} \)
17 \( 1 - 5.12e5T + 1.18e11T^{2} \)
19 \( 1 - 8.50e5T + 3.22e11T^{2} \)
23 \( 1 - 1.92e6T + 1.80e12T^{2} \)
29 \( 1 + 6.01e5T + 1.45e13T^{2} \)
31 \( 1 + 8.43e5T + 2.64e13T^{2} \)
37 \( 1 - 1.61e7T + 1.29e14T^{2} \)
41 \( 1 - 2.86e7T + 3.27e14T^{2} \)
43 \( 1 - 1.17e7T + 5.02e14T^{2} \)
47 \( 1 - 7.59e5T + 1.11e15T^{2} \)
53 \( 1 + 6.97e7T + 3.29e15T^{2} \)
61 \( 1 - 7.60e7T + 1.16e16T^{2} \)
67 \( 1 + 2.31e8T + 2.72e16T^{2} \)
71 \( 1 + 2.17e7T + 4.58e16T^{2} \)
73 \( 1 - 3.17e8T + 5.88e16T^{2} \)
79 \( 1 - 5.20e7T + 1.19e17T^{2} \)
83 \( 1 + 4.88e8T + 1.86e17T^{2} \)
89 \( 1 - 3.00e8T + 3.50e17T^{2} \)
97 \( 1 - 6.18e7T + 7.60e17T^{2} \)
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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

−11.08136419727497989790874145264, −9.710629663248368143912432584375, −9.260544400367516546975997280657, −7.934183774039183421266329616433, −7.20940384431717389806383427476, −5.82816791249466328034125926526, −4.58460102713239238596514345432, −3.50092625427860649631242747377, −1.30143778380932540730290226139, −0.818277480529402993687567937664, 0.818277480529402993687567937664, 1.30143778380932540730290226139, 3.50092625427860649631242747377, 4.58460102713239238596514345432, 5.82816791249466328034125926526, 7.20940384431717389806383427476, 7.934183774039183421266329616433, 9.260544400367516546975997280657, 9.710629663248368143912432584375, 11.08136419727497989790874145264

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