Properties

Label 2-177-1.1-c7-0-11
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 9.54·2-s + 27·3-s − 36.8·4-s + 248.·5-s − 257.·6-s − 1.45e3·7-s + 1.57e3·8-s + 729·9-s − 2.37e3·10-s − 6.46e3·11-s − 994.·12-s + 4.14e3·13-s + 1.38e4·14-s + 6.72e3·15-s − 1.03e4·16-s − 2.58e4·17-s − 6.96e3·18-s + 2.68e4·19-s − 9.16e3·20-s − 3.92e4·21-s + 6.17e4·22-s + 5.88e4·23-s + 4.24e4·24-s − 1.61e4·25-s − 3.95e4·26-s + 1.96e4·27-s + 5.35e4·28-s + ⋯
L(s)  = 1  − 0.843·2-s + 0.577·3-s − 0.287·4-s + 0.890·5-s − 0.487·6-s − 1.60·7-s + 1.08·8-s + 0.333·9-s − 0.751·10-s − 1.46·11-s − 0.166·12-s + 0.522·13-s + 1.35·14-s + 0.514·15-s − 0.629·16-s − 1.27·17-s − 0.281·18-s + 0.898·19-s − 0.256·20-s − 0.924·21-s + 1.23·22-s + 1.00·23-s + 0.627·24-s − 0.206·25-s − 0.441·26-s + 0.192·27-s + 0.460·28-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

\(L(4)\) \(\approx\) \(1.030927411\)
\(L(\frac12)\) \(\approx\) \(1.030927411\)
\(L(\frac{9}{2})\) 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 \( 1 + 9.54T + 128T^{2} \)
5 \( 1 - 248.T + 7.81e4T^{2} \)
7 \( 1 + 1.45e3T + 8.23e5T^{2} \)
11 \( 1 + 6.46e3T + 1.94e7T^{2} \)
13 \( 1 - 4.14e3T + 6.27e7T^{2} \)
17 \( 1 + 2.58e4T + 4.10e8T^{2} \)
19 \( 1 - 2.68e4T + 8.93e8T^{2} \)
23 \( 1 - 5.88e4T + 3.40e9T^{2} \)
29 \( 1 - 1.37e4T + 1.72e10T^{2} \)
31 \( 1 - 2.58e4T + 2.75e10T^{2} \)
37 \( 1 - 1.24e5T + 9.49e10T^{2} \)
41 \( 1 + 2.01e5T + 1.94e11T^{2} \)
43 \( 1 - 2.46e4T + 2.71e11T^{2} \)
47 \( 1 + 4.12e5T + 5.06e11T^{2} \)
53 \( 1 + 4.35e4T + 1.17e12T^{2} \)
61 \( 1 + 1.61e6T + 3.14e12T^{2} \)
67 \( 1 - 1.37e6T + 6.06e12T^{2} \)
71 \( 1 - 2.87e6T + 9.09e12T^{2} \)
73 \( 1 - 4.37e6T + 1.10e13T^{2} \)
79 \( 1 + 9.60e5T + 1.92e13T^{2} \)
83 \( 1 - 6.21e6T + 2.71e13T^{2} \)
89 \( 1 - 9.66e6T + 4.42e13T^{2} \)
97 \( 1 + 1.18e7T + 8.07e13T^{2} \)
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

−10.88686133444356580053780398657, −9.994849030628605684152572311282, −9.451657459074613867257513909962, −8.632979059953803135349446150776, −7.45826320644479793955532152614, −6.32694449905131090933302014907, −4.98572579857301080168359086632, −3.34802161674116155137835919741, −2.20157614224029634445084867187, −0.59870669636246793189453459867, 0.59870669636246793189453459867, 2.20157614224029634445084867187, 3.34802161674116155137835919741, 4.98572579857301080168359086632, 6.32694449905131090933302014907, 7.45826320644479793955532152614, 8.632979059953803135349446150776, 9.451657459074613867257513909962, 9.994849030628605684152572311282, 10.88686133444356580053780398657

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