Properties

Label 2-177-1.1-c9-0-59
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 24.7·2-s − 81·3-s + 98.6·4-s + 583.·5-s + 2.00e3·6-s + 1.00e4·7-s + 1.02e4·8-s + 6.56e3·9-s − 1.44e4·10-s + 3.33e4·11-s − 7.99e3·12-s − 7.26e4·13-s − 2.48e5·14-s − 4.72e4·15-s − 3.02e5·16-s − 3.36e5·17-s − 1.62e5·18-s + 3.17e5·19-s + 5.75e4·20-s − 8.14e5·21-s − 8.23e5·22-s − 1.35e6·23-s − 8.27e5·24-s − 1.61e6·25-s + 1.79e6·26-s − 5.31e5·27-s + 9.92e5·28-s + ⋯
L(s)  = 1  − 1.09·2-s − 0.577·3-s + 0.192·4-s + 0.417·5-s + 0.630·6-s + 1.58·7-s + 0.881·8-s + 0.333·9-s − 0.455·10-s + 0.686·11-s − 0.111·12-s − 0.705·13-s − 1.72·14-s − 0.241·15-s − 1.15·16-s − 0.976·17-s − 0.364·18-s + 0.558·19-s + 0.0804·20-s − 0.913·21-s − 0.749·22-s − 1.01·23-s − 0.509·24-s − 0.825·25-s + 0.770·26-s − 0.192·27-s + 0.305·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 + 24.7T + 512T^{2} \)
5 \( 1 - 583.T + 1.95e6T^{2} \)
7 \( 1 - 1.00e4T + 4.03e7T^{2} \)
11 \( 1 - 3.33e4T + 2.35e9T^{2} \)
13 \( 1 + 7.26e4T + 1.06e10T^{2} \)
17 \( 1 + 3.36e5T + 1.18e11T^{2} \)
19 \( 1 - 3.17e5T + 3.22e11T^{2} \)
23 \( 1 + 1.35e6T + 1.80e12T^{2} \)
29 \( 1 - 2.66e6T + 1.45e13T^{2} \)
31 \( 1 - 5.69e6T + 2.64e13T^{2} \)
37 \( 1 + 6.09e6T + 1.29e14T^{2} \)
41 \( 1 - 2.11e6T + 3.27e14T^{2} \)
43 \( 1 + 2.57e7T + 5.02e14T^{2} \)
47 \( 1 + 3.88e7T + 1.11e15T^{2} \)
53 \( 1 + 9.44e6T + 3.29e15T^{2} \)
61 \( 1 + 5.09e7T + 1.16e16T^{2} \)
67 \( 1 + 1.19e8T + 2.72e16T^{2} \)
71 \( 1 - 2.67e8T + 4.58e16T^{2} \)
73 \( 1 + 3.11e8T + 5.88e16T^{2} \)
79 \( 1 - 4.14e8T + 1.19e17T^{2} \)
83 \( 1 - 2.74e8T + 1.86e17T^{2} \)
89 \( 1 + 7.08e8T + 3.50e17T^{2} \)
97 \( 1 + 3.69e8T + 7.60e17T^{2} \)
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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

−10.35784077993178417807864023717, −9.577720658243148730678063757732, −8.478096828901995172768600014250, −7.71474705388651954779081959362, −6.53990439373000889147765929576, −5.09146836232634138683790868638, −4.33981688116566754688337806528, −2.02332794436260506577758763518, −1.27253348674204483617660028565, 0, 1.27253348674204483617660028565, 2.02332794436260506577758763518, 4.33981688116566754688337806528, 5.09146836232634138683790868638, 6.53990439373000889147765929576, 7.71474705388651954779081959362, 8.478096828901995172768600014250, 9.577720658243148730678063757732, 10.35784077993178417807864023717

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