Properties

Label 2-177-1.1-c13-0-102
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  + 149.·2-s + 729·3-s + 1.41e4·4-s + 6.31e4·5-s + 1.08e5·6-s + 1.47e5·7-s + 8.82e5·8-s + 5.31e5·9-s + 9.42e6·10-s + 7.64e5·11-s + 1.02e7·12-s + 1.16e7·13-s + 2.20e7·14-s + 4.60e7·15-s + 1.62e7·16-s − 1.70e8·17-s + 7.93e7·18-s − 2.91e7·19-s + 8.90e8·20-s + 1.07e8·21-s + 1.14e8·22-s + 2.00e7·23-s + 6.43e8·24-s + 2.76e9·25-s + 1.74e9·26-s + 3.87e8·27-s + 2.08e9·28-s + ⋯
L(s)  = 1  + 1.64·2-s + 0.577·3-s + 1.72·4-s + 1.80·5-s + 0.952·6-s + 0.473·7-s + 1.18·8-s + 0.333·9-s + 2.98·10-s + 0.130·11-s + 0.993·12-s + 0.671·13-s + 0.781·14-s + 1.04·15-s + 0.241·16-s − 1.70·17-s + 0.549·18-s − 0.142·19-s + 3.11·20-s + 0.273·21-s + 0.214·22-s + 0.0281·23-s + 0.687·24-s + 2.26·25-s + 1.10·26-s + 0.192·27-s + 0.815·28-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

\(L(7)\) \(\approx\) \(13.19577701\)
\(L(\frac12)\) \(\approx\) \(13.19577701\)
\(L(\frac{15}{2})\) 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 \( 1 - 149.T + 8.19e3T^{2} \)
5 \( 1 - 6.31e4T + 1.22e9T^{2} \)
7 \( 1 - 1.47e5T + 9.68e10T^{2} \)
11 \( 1 - 7.64e5T + 3.45e13T^{2} \)
13 \( 1 - 1.16e7T + 3.02e14T^{2} \)
17 \( 1 + 1.70e8T + 9.90e15T^{2} \)
19 \( 1 + 2.91e7T + 4.20e16T^{2} \)
23 \( 1 - 2.00e7T + 5.04e17T^{2} \)
29 \( 1 - 4.66e9T + 1.02e19T^{2} \)
31 \( 1 - 6.20e9T + 2.44e19T^{2} \)
37 \( 1 - 2.76e10T + 2.43e20T^{2} \)
41 \( 1 + 2.43e9T + 9.25e20T^{2} \)
43 \( 1 + 3.32e10T + 1.71e21T^{2} \)
47 \( 1 + 4.72e10T + 5.46e21T^{2} \)
53 \( 1 - 5.80e10T + 2.60e22T^{2} \)
61 \( 1 + 2.95e10T + 1.61e23T^{2} \)
67 \( 1 + 9.30e11T + 5.48e23T^{2} \)
71 \( 1 - 1.05e12T + 1.16e24T^{2} \)
73 \( 1 + 2.06e11T + 1.67e24T^{2} \)
79 \( 1 - 1.54e12T + 4.66e24T^{2} \)
83 \( 1 - 2.83e12T + 8.87e24T^{2} \)
89 \( 1 + 3.27e12T + 2.19e25T^{2} \)
97 \( 1 + 5.78e12T + 6.73e25T^{2} \)
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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

−10.51661904157336474693184326595, −9.397497557741108452563436898717, −8.395708571669314592388307981740, −6.57970695990001732572135796912, −6.27080788249483728908937707166, −5.02669001223948394006269485005, −4.32615578309732603720895603164, −2.87638438095437366792503183950, −2.24738513453781160546023267098, −1.31783663745694760513173757896, 1.31783663745694760513173757896, 2.24738513453781160546023267098, 2.87638438095437366792503183950, 4.32615578309732603720895603164, 5.02669001223948394006269485005, 6.27080788249483728908937707166, 6.57970695990001732572135796912, 8.395708571669314592388307981740, 9.397497557741108452563436898717, 10.51661904157336474693184326595

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