Properties

Label 2-177-1.1-c7-0-49
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 $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 16.2·2-s − 27·3-s + 137.·4-s + 495.·5-s + 439.·6-s + 565.·7-s − 155.·8-s + 729·9-s − 8.07e3·10-s − 826.·11-s − 3.71e3·12-s + 1.14e4·13-s − 9.21e3·14-s − 1.33e4·15-s − 1.50e4·16-s − 2.35e4·17-s − 1.18e4·18-s − 5.35e4·19-s + 6.81e4·20-s − 1.52e4·21-s + 1.34e4·22-s − 8.71e4·23-s + 4.19e3·24-s + 1.67e5·25-s − 1.87e5·26-s − 1.96e4·27-s + 7.77e4·28-s + ⋯
L(s)  = 1  − 1.44·2-s − 0.577·3-s + 1.07·4-s + 1.77·5-s + 0.831·6-s + 0.622·7-s − 0.107·8-s + 0.333·9-s − 2.55·10-s − 0.187·11-s − 0.620·12-s + 1.44·13-s − 0.897·14-s − 1.02·15-s − 0.919·16-s − 1.16·17-s − 0.480·18-s − 1.79·19-s + 1.90·20-s − 0.359·21-s + 0.269·22-s − 1.49·23-s + 0.0619·24-s + 2.14·25-s − 2.08·26-s − 0.192·27-s + 0.669·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + 16.2T + 128T^{2} \)
5 \( 1 - 495.T + 7.81e4T^{2} \)
7 \( 1 - 565.T + 8.23e5T^{2} \)
11 \( 1 + 826.T + 1.94e7T^{2} \)
13 \( 1 - 1.14e4T + 6.27e7T^{2} \)
17 \( 1 + 2.35e4T + 4.10e8T^{2} \)
19 \( 1 + 5.35e4T + 8.93e8T^{2} \)
23 \( 1 + 8.71e4T + 3.40e9T^{2} \)
29 \( 1 - 2.67e4T + 1.72e10T^{2} \)
31 \( 1 + 2.31e5T + 2.75e10T^{2} \)
37 \( 1 - 4.09e5T + 9.49e10T^{2} \)
41 \( 1 - 1.89e5T + 1.94e11T^{2} \)
43 \( 1 + 8.49e5T + 2.71e11T^{2} \)
47 \( 1 + 9.02e5T + 5.06e11T^{2} \)
53 \( 1 + 3.24e5T + 1.17e12T^{2} \)
61 \( 1 + 1.58e6T + 3.14e12T^{2} \)
67 \( 1 + 2.57e6T + 6.06e12T^{2} \)
71 \( 1 + 6.78e5T + 9.09e12T^{2} \)
73 \( 1 - 2.50e6T + 1.10e13T^{2} \)
79 \( 1 + 7.33e6T + 1.92e13T^{2} \)
83 \( 1 - 3.91e6T + 2.71e13T^{2} \)
89 \( 1 - 3.80e6T + 4.42e13T^{2} \)
97 \( 1 + 7.68e6T + 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.71577717107506163689617564529, −9.894124585638168203690495988304, −8.936847339772361664005457659053, −8.188877851883826676602129740422, −6.57672581973585852182304531016, −6.00116704055000672176938078802, −4.53399549832963243196248361260, −2.04318667862296378303533185899, −1.53578013912109900291578162600, 0, 1.53578013912109900291578162600, 2.04318667862296378303533185899, 4.53399549832963243196248361260, 6.00116704055000672176938078802, 6.57672581973585852182304531016, 8.188877851883826676602129740422, 8.936847339772361664005457659053, 9.894124585638168203690495988304, 10.71577717107506163689617564529

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