Properties

Label 2-177-1.1-c9-0-17
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  − 39.7·2-s + 81·3-s + 1.06e3·4-s − 1.87e3·5-s − 3.22e3·6-s − 43.1·7-s − 2.21e4·8-s + 6.56e3·9-s + 7.44e4·10-s + 7.61e4·11-s + 8.65e4·12-s − 4.37e4·13-s + 1.71e3·14-s − 1.51e5·15-s + 3.32e5·16-s + 4.20e5·17-s − 2.60e5·18-s − 3.33e5·19-s − 1.99e6·20-s − 3.49e3·21-s − 3.02e6·22-s − 1.14e5·23-s − 1.79e6·24-s + 1.54e6·25-s + 1.73e6·26-s + 5.31e5·27-s − 4.60e4·28-s + ⋯
L(s)  = 1  − 1.75·2-s + 0.577·3-s + 2.08·4-s − 1.33·5-s − 1.01·6-s − 0.00678·7-s − 1.91·8-s + 0.333·9-s + 2.35·10-s + 1.56·11-s + 1.20·12-s − 0.424·13-s + 0.0119·14-s − 0.773·15-s + 1.26·16-s + 1.22·17-s − 0.585·18-s − 0.587·19-s − 2.79·20-s − 0.00391·21-s − 2.75·22-s − 0.0853·23-s − 1.10·24-s + 0.793·25-s + 0.746·26-s + 0.192·27-s − 0.0141·28-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

\(L(5)\) \(\approx\) \(0.8494093527\)
\(L(\frac12)\) \(\approx\) \(0.8494093527\)
\(L(\frac{11}{2})\) 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 \( 1 + 39.7T + 512T^{2} \)
5 \( 1 + 1.87e3T + 1.95e6T^{2} \)
7 \( 1 + 43.1T + 4.03e7T^{2} \)
11 \( 1 - 7.61e4T + 2.35e9T^{2} \)
13 \( 1 + 4.37e4T + 1.06e10T^{2} \)
17 \( 1 - 4.20e5T + 1.18e11T^{2} \)
19 \( 1 + 3.33e5T + 3.22e11T^{2} \)
23 \( 1 + 1.14e5T + 1.80e12T^{2} \)
29 \( 1 - 2.58e6T + 1.45e13T^{2} \)
31 \( 1 + 1.72e6T + 2.64e13T^{2} \)
37 \( 1 - 4.90e6T + 1.29e14T^{2} \)
41 \( 1 - 7.23e6T + 3.27e14T^{2} \)
43 \( 1 + 8.98e6T + 5.02e14T^{2} \)
47 \( 1 - 5.61e7T + 1.11e15T^{2} \)
53 \( 1 + 9.20e7T + 3.29e15T^{2} \)
61 \( 1 + 8.95e7T + 1.16e16T^{2} \)
67 \( 1 + 1.59e8T + 2.72e16T^{2} \)
71 \( 1 - 1.68e8T + 4.58e16T^{2} \)
73 \( 1 + 1.57e8T + 5.88e16T^{2} \)
79 \( 1 - 5.77e8T + 1.19e17T^{2} \)
83 \( 1 + 4.29e8T + 1.86e17T^{2} \)
89 \( 1 + 6.09e8T + 3.50e17T^{2} \)
97 \( 1 - 1.42e9T + 7.60e17T^{2} \)
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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

−10.81275791545625444082402441068, −9.713545204518505802420440647856, −8.953210684785524302527711126009, −8.054434034755585921028361575639, −7.44091309717110451728852711432, −6.44150576637834649583393486641, −4.22970260743883712814296867937, −3.09468037411980301356373069132, −1.59831685085332918171660354573, −0.60328892023246022088976489070, 0.60328892023246022088976489070, 1.59831685085332918171660354573, 3.09468037411980301356373069132, 4.22970260743883712814296867937, 6.44150576637834649583393486641, 7.44091309717110451728852711432, 8.054434034755585921028361575639, 8.953210684785524302527711126009, 9.713545204518505802420440647856, 10.81275791545625444082402441068

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