Properties

Label 2-177-1.1-c9-0-82
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  + 37.7·2-s + 81·3-s + 909.·4-s − 1.22e3·5-s + 3.05e3·6-s − 1.96e3·7-s + 1.50e4·8-s + 6.56e3·9-s − 4.63e4·10-s − 5.48e4·11-s + 7.37e4·12-s + 3.43e4·13-s − 7.42e4·14-s − 9.95e4·15-s + 9.99e4·16-s − 1.22e5·17-s + 2.47e5·18-s + 8.38e5·19-s − 1.11e6·20-s − 1.59e5·21-s − 2.06e6·22-s − 2.32e6·23-s + 1.21e6·24-s − 4.41e5·25-s + 1.29e6·26-s + 5.31e5·27-s − 1.79e6·28-s + ⋯
L(s)  = 1  + 1.66·2-s + 0.577·3-s + 1.77·4-s − 0.879·5-s + 0.962·6-s − 0.309·7-s + 1.29·8-s + 0.333·9-s − 1.46·10-s − 1.12·11-s + 1.02·12-s + 0.333·13-s − 0.516·14-s − 0.507·15-s + 0.381·16-s − 0.356·17-s + 0.555·18-s + 1.47·19-s − 1.56·20-s − 0.178·21-s − 1.88·22-s − 1.73·23-s + 0.747·24-s − 0.225·25-s + 0.555·26-s + 0.192·27-s − 0.550·28-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 \( 1 - 37.7T + 512T^{2} \)
5 \( 1 + 1.22e3T + 1.95e6T^{2} \)
7 \( 1 + 1.96e3T + 4.03e7T^{2} \)
11 \( 1 + 5.48e4T + 2.35e9T^{2} \)
13 \( 1 - 3.43e4T + 1.06e10T^{2} \)
17 \( 1 + 1.22e5T + 1.18e11T^{2} \)
19 \( 1 - 8.38e5T + 3.22e11T^{2} \)
23 \( 1 + 2.32e6T + 1.80e12T^{2} \)
29 \( 1 + 6.13e6T + 1.45e13T^{2} \)
31 \( 1 - 3.50e6T + 2.64e13T^{2} \)
37 \( 1 + 9.25e6T + 1.29e14T^{2} \)
41 \( 1 + 1.75e7T + 3.27e14T^{2} \)
43 \( 1 - 2.04e7T + 5.02e14T^{2} \)
47 \( 1 - 1.13e7T + 1.11e15T^{2} \)
53 \( 1 + 5.64e7T + 3.29e15T^{2} \)
61 \( 1 + 1.93e8T + 1.16e16T^{2} \)
67 \( 1 - 2.97e8T + 2.72e16T^{2} \)
71 \( 1 + 3.55e8T + 4.58e16T^{2} \)
73 \( 1 - 4.51e8T + 5.88e16T^{2} \)
79 \( 1 - 3.03e8T + 1.19e17T^{2} \)
83 \( 1 - 6.28e8T + 1.86e17T^{2} \)
89 \( 1 + 8.08e8T + 3.50e17T^{2} \)
97 \( 1 - 5.73e8T + 7.60e17T^{2} \)
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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

−10.92006461535227399904316662265, −9.624513323844446097793181454769, −8.075256516304117576214338610640, −7.34203932809597781298772380536, −6.03231922724187157443968518002, −4.98380478608367191414558407276, −3.85075528766264004092402925825, −3.21339261019040234302251855740, −2.02925738877814805117914393762, 0, 2.02925738877814805117914393762, 3.21339261019040234302251855740, 3.85075528766264004092402925825, 4.98380478608367191414558407276, 6.03231922724187157443968518002, 7.34203932809597781298772380536, 8.075256516304117576214338610640, 9.624513323844446097793181454769, 10.92006461535227399904316662265

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