Properties

Label 2-177-1.1-c11-0-70
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.91·2-s + 243·3-s − 2.02e3·4-s − 1.11e3·5-s − 1.19e3·6-s − 2.14e4·7-s + 2.00e4·8-s + 5.90e4·9-s + 5.49e3·10-s + 1.10e5·11-s − 4.91e5·12-s − 3.43e4·13-s + 1.05e5·14-s − 2.71e5·15-s + 4.04e6·16-s − 8.45e6·17-s − 2.90e5·18-s + 1.50e7·19-s + 2.26e6·20-s − 5.20e6·21-s − 5.41e5·22-s + 7.31e6·23-s + 4.86e6·24-s − 4.75e7·25-s + 1.68e5·26-s + 1.43e7·27-s + 4.33e7·28-s + ⋯
L(s)  = 1  − 0.108·2-s + 0.577·3-s − 0.988·4-s − 0.159·5-s − 0.0627·6-s − 0.481·7-s + 0.215·8-s + 0.333·9-s + 0.0173·10-s + 0.206·11-s − 0.570·12-s − 0.0256·13-s + 0.0523·14-s − 0.0923·15-s + 0.964·16-s − 1.44·17-s − 0.0362·18-s + 1.39·19-s + 0.158·20-s − 0.278·21-s − 0.0224·22-s + 0.236·23-s + 0.124·24-s − 0.974·25-s + 0.00278·26-s + 0.192·27-s + 0.475·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 243T \)
59 \( 1 + 7.14e8T \)
good2 \( 1 + 4.91T + 2.04e3T^{2} \)
5 \( 1 + 1.11e3T + 4.88e7T^{2} \)
7 \( 1 + 2.14e4T + 1.97e9T^{2} \)
11 \( 1 - 1.10e5T + 2.85e11T^{2} \)
13 \( 1 + 3.43e4T + 1.79e12T^{2} \)
17 \( 1 + 8.45e6T + 3.42e13T^{2} \)
19 \( 1 - 1.50e7T + 1.16e14T^{2} \)
23 \( 1 - 7.31e6T + 9.52e14T^{2} \)
29 \( 1 + 3.39e7T + 1.22e16T^{2} \)
31 \( 1 - 1.38e8T + 2.54e16T^{2} \)
37 \( 1 - 5.04e8T + 1.77e17T^{2} \)
41 \( 1 - 6.49e8T + 5.50e17T^{2} \)
43 \( 1 - 1.25e9T + 9.29e17T^{2} \)
47 \( 1 + 1.92e9T + 2.47e18T^{2} \)
53 \( 1 - 5.25e9T + 9.26e18T^{2} \)
61 \( 1 + 8.19e9T + 4.35e19T^{2} \)
67 \( 1 + 8.86e9T + 1.22e20T^{2} \)
71 \( 1 + 3.46e9T + 2.31e20T^{2} \)
73 \( 1 - 5.04e9T + 3.13e20T^{2} \)
79 \( 1 - 2.53e10T + 7.47e20T^{2} \)
83 \( 1 - 2.79e10T + 1.28e21T^{2} \)
89 \( 1 + 9.94e10T + 2.77e21T^{2} \)
97 \( 1 + 1.07e11T + 7.15e21T^{2} \)
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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

−9.738380786014053487474576494118, −9.297791313847486020905854936268, −8.269059949410790864833666976718, −7.32552823642821139223271006192, −5.99173983581204080049109437373, −4.65683233204665785751561380377, −3.80071989601146801537384444886, −2.65405460579938918716553734062, −1.14372295037178094476659407697, 0, 1.14372295037178094476659407697, 2.65405460579938918716553734062, 3.80071989601146801537384444886, 4.65683233204665785751561380377, 5.99173983581204080049109437373, 7.32552823642821139223271006192, 8.269059949410790864833666976718, 9.297791313847486020905854936268, 9.738380786014053487474576494118

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