Properties

Label 2-177-1.1-c9-0-50
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  + 1.56·2-s + 81·3-s − 509.·4-s − 1.26e3·5-s + 126.·6-s − 7.65e3·7-s − 1.60e3·8-s + 6.56e3·9-s − 1.97e3·10-s + 7.03e4·11-s − 4.12e4·12-s − 4.76e4·13-s − 1.19e4·14-s − 1.02e5·15-s + 2.58e5·16-s + 3.89e5·17-s + 1.02e4·18-s + 5.11e5·19-s + 6.43e5·20-s − 6.20e5·21-s + 1.10e5·22-s + 1.72e5·23-s − 1.29e5·24-s − 3.58e5·25-s − 7.46e4·26-s + 5.31e5·27-s + 3.90e6·28-s + ⋯
L(s)  = 1  + 0.0692·2-s + 0.577·3-s − 0.995·4-s − 0.903·5-s + 0.0399·6-s − 1.20·7-s − 0.138·8-s + 0.333·9-s − 0.0625·10-s + 1.44·11-s − 0.574·12-s − 0.463·13-s − 0.0834·14-s − 0.521·15-s + 0.985·16-s + 1.13·17-s + 0.0230·18-s + 0.900·19-s + 0.899·20-s − 0.695·21-s + 0.100·22-s + 0.128·23-s − 0.0797·24-s − 0.183·25-s − 0.0320·26-s + 0.192·27-s + 1.19·28-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 \( 1 - 1.56T + 512T^{2} \)
5 \( 1 + 1.26e3T + 1.95e6T^{2} \)
7 \( 1 + 7.65e3T + 4.03e7T^{2} \)
11 \( 1 - 7.03e4T + 2.35e9T^{2} \)
13 \( 1 + 4.76e4T + 1.06e10T^{2} \)
17 \( 1 - 3.89e5T + 1.18e11T^{2} \)
19 \( 1 - 5.11e5T + 3.22e11T^{2} \)
23 \( 1 - 1.72e5T + 1.80e12T^{2} \)
29 \( 1 - 4.84e6T + 1.45e13T^{2} \)
31 \( 1 + 1.87e5T + 2.64e13T^{2} \)
37 \( 1 + 2.06e7T + 1.29e14T^{2} \)
41 \( 1 - 1.23e7T + 3.27e14T^{2} \)
43 \( 1 + 3.93e7T + 5.02e14T^{2} \)
47 \( 1 + 3.33e7T + 1.11e15T^{2} \)
53 \( 1 - 1.91e7T + 3.29e15T^{2} \)
61 \( 1 + 1.21e8T + 1.16e16T^{2} \)
67 \( 1 - 2.99e6T + 2.72e16T^{2} \)
71 \( 1 + 1.08e8T + 4.58e16T^{2} \)
73 \( 1 - 7.68e7T + 5.88e16T^{2} \)
79 \( 1 - 3.66e8T + 1.19e17T^{2} \)
83 \( 1 - 4.97e8T + 1.86e17T^{2} \)
89 \( 1 - 4.74e8T + 3.50e17T^{2} \)
97 \( 1 + 3.38e8T + 7.60e17T^{2} \)
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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

−10.10142803402548107049164003466, −9.488502741261435835439067928529, −8.599090168208643902781968405207, −7.55804679715634307358557024481, −6.44670102214275231378531546136, −4.93905328504460000845609328276, −3.67048252348304748837499773653, −3.29653158525177859601862824887, −1.15883047383314893714019520627, 0, 1.15883047383314893714019520627, 3.29653158525177859601862824887, 3.67048252348304748837499773653, 4.93905328504460000845609328276, 6.44670102214275231378531546136, 7.55804679715634307358557024481, 8.599090168208643902781968405207, 9.488502741261435835439067928529, 10.10142803402548107049164003466

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