Properties

Label 2-177-1.1-c7-0-53
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.6·2-s + 27·3-s + 88.0·4-s + 457.·5-s + 396.·6-s + 1.57e3·7-s − 587.·8-s + 729·9-s + 6.72e3·10-s + 6.84e3·11-s + 2.37e3·12-s + 4.11e3·13-s + 2.31e4·14-s + 1.23e4·15-s − 1.99e4·16-s − 3.48e4·17-s + 1.07e4·18-s − 3.08e4·19-s + 4.02e4·20-s + 4.25e4·21-s + 1.00e5·22-s − 5.14e4·23-s − 1.58e4·24-s + 1.31e5·25-s + 6.05e4·26-s + 1.96e4·27-s + 1.38e5·28-s + ⋯
L(s)  = 1  + 1.29·2-s + 0.577·3-s + 0.687·4-s + 1.63·5-s + 0.750·6-s + 1.73·7-s − 0.405·8-s + 0.333·9-s + 2.12·10-s + 1.54·11-s + 0.396·12-s + 0.519·13-s + 2.25·14-s + 0.945·15-s − 1.21·16-s − 1.72·17-s + 0.433·18-s − 1.03·19-s + 1.12·20-s + 1.00·21-s + 2.01·22-s − 0.881·23-s − 0.234·24-s + 1.68·25-s + 0.675·26-s + 0.192·27-s + 1.19·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(8.145858552\)
\(L(\frac12)\) \(\approx\) \(8.145858552\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 27T \)
59 \( 1 - 2.05e5T \)
good2 \( 1 - 14.6T + 128T^{2} \)
5 \( 1 - 457.T + 7.81e4T^{2} \)
7 \( 1 - 1.57e3T + 8.23e5T^{2} \)
11 \( 1 - 6.84e3T + 1.94e7T^{2} \)
13 \( 1 - 4.11e3T + 6.27e7T^{2} \)
17 \( 1 + 3.48e4T + 4.10e8T^{2} \)
19 \( 1 + 3.08e4T + 8.93e8T^{2} \)
23 \( 1 + 5.14e4T + 3.40e9T^{2} \)
29 \( 1 - 1.06e5T + 1.72e10T^{2} \)
31 \( 1 + 2.36e5T + 2.75e10T^{2} \)
37 \( 1 + 3.20e5T + 9.49e10T^{2} \)
41 \( 1 + 2.27e5T + 1.94e11T^{2} \)
43 \( 1 - 1.56e5T + 2.71e11T^{2} \)
47 \( 1 - 9.02e5T + 5.06e11T^{2} \)
53 \( 1 + 1.28e6T + 1.17e12T^{2} \)
61 \( 1 - 7.19e5T + 3.14e12T^{2} \)
67 \( 1 + 2.73e6T + 6.06e12T^{2} \)
71 \( 1 - 2.81e6T + 9.09e12T^{2} \)
73 \( 1 + 8.80e5T + 1.10e13T^{2} \)
79 \( 1 - 2.91e6T + 1.92e13T^{2} \)
83 \( 1 + 6.61e6T + 2.71e13T^{2} \)
89 \( 1 + 6.36e6T + 4.42e13T^{2} \)
97 \( 1 - 1.73e7T + 8.07e13T^{2} \)
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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

−11.55636368718697059510262248814, −10.62829008208414114946046319328, −9.085220717896676645996982349151, −8.672293455303568099444019799056, −6.76006081528733614072739517651, −5.94178058615290036836070709466, −4.77685499325025087840456690911, −3.97829978531049759646803739438, −2.19510566880617069574339148028, −1.64908185379076646575161132804, 1.64908185379076646575161132804, 2.19510566880617069574339148028, 3.97829978531049759646803739438, 4.77685499325025087840456690911, 5.94178058615290036836070709466, 6.76006081528733614072739517651, 8.672293455303568099444019799056, 9.085220717896676645996982349151, 10.62829008208414114946046319328, 11.55636368718697059510262248814

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