Properties

Label 2-177-1.1-c13-0-70
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 50.5·2-s + 729·3-s − 5.63e3·4-s + 3.74e4·5-s − 3.68e4·6-s + 1.31e5·7-s + 6.99e5·8-s + 5.31e5·9-s − 1.89e6·10-s + 4.83e6·11-s − 4.10e6·12-s + 7.84e6·13-s − 6.64e6·14-s + 2.73e7·15-s + 1.08e7·16-s + 1.27e8·17-s − 2.68e7·18-s + 1.07e8·19-s − 2.11e8·20-s + 9.58e7·21-s − 2.44e8·22-s + 4.53e8·23-s + 5.09e8·24-s + 1.83e8·25-s − 3.96e8·26-s + 3.87e8·27-s − 7.41e8·28-s + ⋯
L(s)  = 1  − 0.558·2-s + 0.577·3-s − 0.688·4-s + 1.07·5-s − 0.322·6-s + 0.422·7-s + 0.942·8-s + 0.333·9-s − 0.599·10-s + 0.823·11-s − 0.397·12-s + 0.450·13-s − 0.235·14-s + 0.619·15-s + 0.161·16-s + 1.28·17-s − 0.186·18-s + 0.526·19-s − 0.738·20-s + 0.243·21-s − 0.459·22-s + 0.638·23-s + 0.544·24-s + 0.150·25-s − 0.251·26-s + 0.192·27-s − 0.290·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(3.346239833\)
\(L(\frac12)\) \(\approx\) \(3.346239833\)
\(L(\frac{15}{2})\) 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 \( 1 + 50.5T + 8.19e3T^{2} \)
5 \( 1 - 3.74e4T + 1.22e9T^{2} \)
7 \( 1 - 1.31e5T + 9.68e10T^{2} \)
11 \( 1 - 4.83e6T + 3.45e13T^{2} \)
13 \( 1 - 7.84e6T + 3.02e14T^{2} \)
17 \( 1 - 1.27e8T + 9.90e15T^{2} \)
19 \( 1 - 1.07e8T + 4.20e16T^{2} \)
23 \( 1 - 4.53e8T + 5.04e17T^{2} \)
29 \( 1 - 2.96e9T + 1.02e19T^{2} \)
31 \( 1 - 5.30e9T + 2.44e19T^{2} \)
37 \( 1 - 1.26e10T + 2.43e20T^{2} \)
41 \( 1 - 4.56e10T + 9.25e20T^{2} \)
43 \( 1 - 2.06e10T + 1.71e21T^{2} \)
47 \( 1 + 3.67e10T + 5.46e21T^{2} \)
53 \( 1 + 1.16e11T + 2.60e22T^{2} \)
61 \( 1 + 2.48e11T + 1.61e23T^{2} \)
67 \( 1 - 7.76e11T + 5.48e23T^{2} \)
71 \( 1 - 6.93e9T + 1.16e24T^{2} \)
73 \( 1 - 1.40e12T + 1.67e24T^{2} \)
79 \( 1 + 1.00e12T + 4.66e24T^{2} \)
83 \( 1 + 3.96e12T + 8.87e24T^{2} \)
89 \( 1 + 3.15e12T + 2.19e25T^{2} \)
97 \( 1 + 3.57e12T + 6.73e25T^{2} \)
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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

−9.834703590146575928546565479863, −9.505726843500138730341415625976, −8.480742414037217758130211611893, −7.67711983650930485486527339018, −6.30286342758068724236473173181, −5.18232797015386100819961283305, −4.09570180108831347271206865327, −2.83890459352242005154179878005, −1.43306229773438646734449471651, −0.984801374974415170397633014926, 0.984801374974415170397633014926, 1.43306229773438646734449471651, 2.83890459352242005154179878005, 4.09570180108831347271206865327, 5.18232797015386100819961283305, 6.30286342758068724236473173181, 7.67711983650930485486527339018, 8.480742414037217758130211611893, 9.505726843500138730341415625976, 9.834703590146575928546565479863

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