Properties

Label 2-177-1.1-c13-0-107
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 31.8·2-s + 729·3-s − 7.17e3·4-s + 6.11e4·5-s + 2.32e4·6-s − 2.36e5·7-s − 4.90e5·8-s + 5.31e5·9-s + 1.94e6·10-s + 1.22e6·11-s − 5.23e6·12-s − 6.08e6·13-s − 7.55e6·14-s + 4.45e7·15-s + 4.31e7·16-s + 1.15e8·17-s + 1.69e7·18-s − 2.61e8·19-s − 4.38e8·20-s − 1.72e8·21-s + 3.91e7·22-s − 1.17e9·23-s − 3.57e8·24-s + 2.51e9·25-s − 1.94e8·26-s + 3.87e8·27-s + 1.69e9·28-s + ⋯
L(s)  = 1  + 0.352·2-s + 0.577·3-s − 0.875·4-s + 1.74·5-s + 0.203·6-s − 0.760·7-s − 0.661·8-s + 0.333·9-s + 0.616·10-s + 0.208·11-s − 0.505·12-s − 0.349·13-s − 0.268·14-s + 1.00·15-s + 0.642·16-s + 1.16·17-s + 0.117·18-s − 1.27·19-s − 1.53·20-s − 0.439·21-s + 0.0736·22-s − 1.65·23-s − 0.381·24-s + 2.05·25-s − 0.123·26-s + 0.192·27-s + 0.666·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 729T \)
59 \( 1 - 4.21e10T \)
good2 \( 1 - 31.8T + 8.19e3T^{2} \)
5 \( 1 - 6.11e4T + 1.22e9T^{2} \)
7 \( 1 + 2.36e5T + 9.68e10T^{2} \)
11 \( 1 - 1.22e6T + 3.45e13T^{2} \)
13 \( 1 + 6.08e6T + 3.02e14T^{2} \)
17 \( 1 - 1.15e8T + 9.90e15T^{2} \)
19 \( 1 + 2.61e8T + 4.20e16T^{2} \)
23 \( 1 + 1.17e9T + 5.04e17T^{2} \)
29 \( 1 + 1.96e8T + 1.02e19T^{2} \)
31 \( 1 - 1.49e9T + 2.44e19T^{2} \)
37 \( 1 - 1.48e10T + 2.43e20T^{2} \)
41 \( 1 + 1.03e10T + 9.25e20T^{2} \)
43 \( 1 + 3.86e10T + 1.71e21T^{2} \)
47 \( 1 + 1.59e10T + 5.46e21T^{2} \)
53 \( 1 + 3.65e10T + 2.60e22T^{2} \)
61 \( 1 + 7.27e11T + 1.61e23T^{2} \)
67 \( 1 - 2.38e11T + 5.48e23T^{2} \)
71 \( 1 + 4.31e11T + 1.16e24T^{2} \)
73 \( 1 + 8.74e11T + 1.67e24T^{2} \)
79 \( 1 - 1.86e12T + 4.66e24T^{2} \)
83 \( 1 - 2.77e11T + 8.87e24T^{2} \)
89 \( 1 + 1.37e12T + 2.19e25T^{2} \)
97 \( 1 - 6.08e12T + 6.73e25T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.828116935741241310553556806999, −9.103747649626448875849363745178, −8.059332020008393674274547558403, −6.41291891659918246496503725156, −5.82578346873924582502288526220, −4.67022782957587813396745010801, −3.48437193216640861978077173579, −2.45984487869027760632288822188, −1.41537570539904098738456277699, 0, 1.41537570539904098738456277699, 2.45984487869027760632288822188, 3.48437193216640861978077173579, 4.67022782957587813396745010801, 5.82578346873924582502288526220, 6.41291891659918246496503725156, 8.059332020008393674274547558403, 9.103747649626448875849363745178, 9.828116935741241310553556806999

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