Properties

Label 2-177-1.1-c13-0-41
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  − 102.·2-s + 729·3-s + 2.27e3·4-s − 9.86e3·5-s − 7.45e4·6-s + 4.39e5·7-s + 6.05e5·8-s + 5.31e5·9-s + 1.00e6·10-s − 9.59e5·11-s + 1.65e6·12-s + 5.60e6·13-s − 4.50e7·14-s − 7.18e6·15-s − 8.05e7·16-s − 1.88e8·17-s − 5.43e7·18-s − 3.80e6·19-s − 2.24e7·20-s + 3.20e8·21-s + 9.81e7·22-s + 1.27e9·23-s + 4.41e8·24-s − 1.12e9·25-s − 5.73e8·26-s + 3.87e8·27-s + 1.00e9·28-s + ⋯
L(s)  = 1  − 1.13·2-s + 0.577·3-s + 0.277·4-s − 0.282·5-s − 0.652·6-s + 1.41·7-s + 0.816·8-s + 0.333·9-s + 0.319·10-s − 0.163·11-s + 0.160·12-s + 0.322·13-s − 1.59·14-s − 0.162·15-s − 1.20·16-s − 1.89·17-s − 0.376·18-s − 0.0185·19-s − 0.0784·20-s + 0.815·21-s + 0.184·22-s + 1.79·23-s + 0.471·24-s − 0.920·25-s − 0.364·26-s + 0.192·27-s + 0.392·28-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

\(L(7)\) \(\approx\) \(1.662909240\)
\(L(\frac12)\) \(\approx\) \(1.662909240\)
\(L(\frac{15}{2})\) 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 \( 1 + 102.T + 8.19e3T^{2} \)
5 \( 1 + 9.86e3T + 1.22e9T^{2} \)
7 \( 1 - 4.39e5T + 9.68e10T^{2} \)
11 \( 1 + 9.59e5T + 3.45e13T^{2} \)
13 \( 1 - 5.60e6T + 3.02e14T^{2} \)
17 \( 1 + 1.88e8T + 9.90e15T^{2} \)
19 \( 1 + 3.80e6T + 4.20e16T^{2} \)
23 \( 1 - 1.27e9T + 5.04e17T^{2} \)
29 \( 1 - 4.95e9T + 1.02e19T^{2} \)
31 \( 1 - 8.24e9T + 2.44e19T^{2} \)
37 \( 1 + 2.50e10T + 2.43e20T^{2} \)
41 \( 1 + 3.30e10T + 9.25e20T^{2} \)
43 \( 1 - 4.90e10T + 1.71e21T^{2} \)
47 \( 1 - 8.62e10T + 5.46e21T^{2} \)
53 \( 1 - 2.96e11T + 2.60e22T^{2} \)
61 \( 1 + 6.27e11T + 1.61e23T^{2} \)
67 \( 1 + 6.94e11T + 5.48e23T^{2} \)
71 \( 1 - 1.94e11T + 1.16e24T^{2} \)
73 \( 1 - 3.28e11T + 1.67e24T^{2} \)
79 \( 1 - 3.01e10T + 4.66e24T^{2} \)
83 \( 1 - 5.53e11T + 8.87e24T^{2} \)
89 \( 1 + 2.05e12T + 2.19e25T^{2} \)
97 \( 1 - 5.32e12T + 6.73e25T^{2} \)
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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

−10.33827451939389125020607118968, −8.842314882884537026471713591943, −8.665953832507744198698594589416, −7.69388366159862837021375085650, −6.78937368074113907489959939478, −4.90831779321139844549419791132, −4.25446340533451014634715702369, −2.56705003412517147765057149546, −1.57671998613038526132747857337, −0.67257818921093984872470953401, 0.67257818921093984872470953401, 1.57671998613038526132747857337, 2.56705003412517147765057149546, 4.25446340533451014634715702369, 4.90831779321139844549419791132, 6.78937368074113907489959939478, 7.69388366159862837021375085650, 8.665953832507744198698594589416, 8.842314882884537026471713591943, 10.33827451939389125020607118968

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