Properties

Label 2-177-1.1-c9-0-71
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  + 33.2·2-s − 81·3-s + 595.·4-s − 1.22e3·5-s − 2.69e3·6-s + 7.34e3·7-s + 2.76e3·8-s + 6.56e3·9-s − 4.06e4·10-s − 2.98e4·11-s − 4.82e4·12-s − 2.27e4·13-s + 2.44e5·14-s + 9.89e4·15-s − 2.12e5·16-s + 6.33e5·17-s + 2.18e5·18-s + 9.43e5·19-s − 7.27e5·20-s − 5.94e5·21-s − 9.92e5·22-s − 2.11e6·23-s − 2.24e5·24-s − 4.59e5·25-s − 7.58e5·26-s − 5.31e5·27-s + 4.37e6·28-s + ⋯
L(s)  = 1  + 1.47·2-s − 0.577·3-s + 1.16·4-s − 0.874·5-s − 0.849·6-s + 1.15·7-s + 0.238·8-s + 0.333·9-s − 1.28·10-s − 0.614·11-s − 0.671·12-s − 0.221·13-s + 1.69·14-s + 0.504·15-s − 0.811·16-s + 1.84·17-s + 0.490·18-s + 1.66·19-s − 1.01·20-s − 0.667·21-s − 0.903·22-s − 1.57·23-s − 0.137·24-s − 0.235·25-s − 0.325·26-s − 0.192·27-s + 1.34·28-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 \( 1 - 33.2T + 512T^{2} \)
5 \( 1 + 1.22e3T + 1.95e6T^{2} \)
7 \( 1 - 7.34e3T + 4.03e7T^{2} \)
11 \( 1 + 2.98e4T + 2.35e9T^{2} \)
13 \( 1 + 2.27e4T + 1.06e10T^{2} \)
17 \( 1 - 6.33e5T + 1.18e11T^{2} \)
19 \( 1 - 9.43e5T + 3.22e11T^{2} \)
23 \( 1 + 2.11e6T + 1.80e12T^{2} \)
29 \( 1 + 3.29e6T + 1.45e13T^{2} \)
31 \( 1 + 5.35e6T + 2.64e13T^{2} \)
37 \( 1 - 1.22e6T + 1.29e14T^{2} \)
41 \( 1 - 1.04e7T + 3.27e14T^{2} \)
43 \( 1 + 2.91e7T + 5.02e14T^{2} \)
47 \( 1 + 2.38e7T + 1.11e15T^{2} \)
53 \( 1 - 4.57e7T + 3.29e15T^{2} \)
61 \( 1 + 1.05e8T + 1.16e16T^{2} \)
67 \( 1 + 3.01e8T + 2.72e16T^{2} \)
71 \( 1 - 6.25e7T + 4.58e16T^{2} \)
73 \( 1 - 1.58e8T + 5.88e16T^{2} \)
79 \( 1 + 4.83e8T + 1.19e17T^{2} \)
83 \( 1 - 2.41e7T + 1.86e17T^{2} \)
89 \( 1 - 3.08e7T + 3.50e17T^{2} \)
97 \( 1 + 1.35e9T + 7.60e17T^{2} \)
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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

−11.07127190296989678274503053799, −9.785751629147977624003503135439, −7.950453714780400564897424639339, −7.41205364293774607562223346664, −5.70035279418871975260159318887, −5.23282924805096438611017345966, −4.14375902701031013661528840748, −3.21706123391003036261334916996, −1.59125703901364919259320940394, 0, 1.59125703901364919259320940394, 3.21706123391003036261334916996, 4.14375902701031013661528840748, 5.23282924805096438611017345966, 5.70035279418871975260159318887, 7.41205364293774607562223346664, 7.950453714780400564897424639339, 9.785751629147977624003503135439, 11.07127190296989678274503053799

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