Properties

Label 2-177-1.1-c13-0-51
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  − 166.·2-s + 729·3-s + 1.96e4·4-s + 4.35e4·5-s − 1.21e5·6-s − 3.95e5·7-s − 1.90e6·8-s + 5.31e5·9-s − 7.26e6·10-s + 2.80e6·11-s + 1.42e7·12-s + 2.66e7·13-s + 6.58e7·14-s + 3.17e7·15-s + 1.56e8·16-s + 1.05e8·17-s − 8.85e7·18-s + 1.56e8·19-s + 8.53e8·20-s − 2.88e8·21-s − 4.67e8·22-s − 2.25e8·23-s − 1.38e9·24-s + 6.76e8·25-s − 4.43e9·26-s + 3.87e8·27-s − 7.74e9·28-s + ⋯
L(s)  = 1  − 1.84·2-s + 0.577·3-s + 2.39·4-s + 1.24·5-s − 1.06·6-s − 1.26·7-s − 2.56·8-s + 0.333·9-s − 2.29·10-s + 0.477·11-s + 1.38·12-s + 1.53·13-s + 2.33·14-s + 0.719·15-s + 2.33·16-s + 1.06·17-s − 0.613·18-s + 0.764·19-s + 2.98·20-s − 0.733·21-s − 0.879·22-s − 0.317·23-s − 1.48·24-s + 0.554·25-s − 2.81·26-s + 0.192·27-s − 3.03·28-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

\(L(7)\) \(\approx\) \(1.815788851\)
\(L(\frac12)\) \(\approx\) \(1.815788851\)
\(L(\frac{15}{2})\) 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 \( 1 + 166.T + 8.19e3T^{2} \)
5 \( 1 - 4.35e4T + 1.22e9T^{2} \)
7 \( 1 + 3.95e5T + 9.68e10T^{2} \)
11 \( 1 - 2.80e6T + 3.45e13T^{2} \)
13 \( 1 - 2.66e7T + 3.02e14T^{2} \)
17 \( 1 - 1.05e8T + 9.90e15T^{2} \)
19 \( 1 - 1.56e8T + 4.20e16T^{2} \)
23 \( 1 + 2.25e8T + 5.04e17T^{2} \)
29 \( 1 - 4.55e8T + 1.02e19T^{2} \)
31 \( 1 - 3.56e9T + 2.44e19T^{2} \)
37 \( 1 - 2.68e10T + 2.43e20T^{2} \)
41 \( 1 + 1.72e10T + 9.25e20T^{2} \)
43 \( 1 - 2.91e10T + 1.71e21T^{2} \)
47 \( 1 - 1.44e8T + 5.46e21T^{2} \)
53 \( 1 - 4.30e10T + 2.60e22T^{2} \)
61 \( 1 + 2.11e11T + 1.61e23T^{2} \)
67 \( 1 + 9.17e11T + 5.48e23T^{2} \)
71 \( 1 - 1.33e12T + 1.16e24T^{2} \)
73 \( 1 + 1.30e12T + 1.67e24T^{2} \)
79 \( 1 - 3.47e12T + 4.66e24T^{2} \)
83 \( 1 - 3.63e12T + 8.87e24T^{2} \)
89 \( 1 + 1.84e12T + 2.19e25T^{2} \)
97 \( 1 + 1.21e13T + 6.73e25T^{2} \)
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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

−9.803838167510711250088156957648, −9.529987215368210380921325025341, −8.653523672240829247836532835891, −7.62369339270210069186483650718, −6.41044132156728585888771327231, −5.98094422633344655458452183350, −3.47571324880959011049501421788, −2.58707152431163994032667916829, −1.44502128276082363891542630219, −0.809526474808124774166722611858, 0.809526474808124774166722611858, 1.44502128276082363891542630219, 2.58707152431163994032667916829, 3.47571324880959011049501421788, 5.98094422633344655458452183350, 6.41044132156728585888771327231, 7.62369339270210069186483650718, 8.653523672240829247836532835891, 9.529987215368210380921325025341, 9.803838167510711250088156957648

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