Properties

Label 2-177-59.58-c4-0-32
Degree $2$
Conductor $177$
Sign $-0.989 - 0.142i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.05i·2-s − 5.19·3-s − 0.408·4-s − 16.4·5-s + 21.0i·6-s + 47.3·7-s − 63.1i·8-s + 27·9-s + 66.4i·10-s − 25.6i·11-s + 2.12·12-s − 105. i·13-s − 191. i·14-s + 85.2·15-s − 262.·16-s + 441.·17-s + ⋯
L(s)  = 1  − 1.01i·2-s − 0.577·3-s − 0.0255·4-s − 0.656·5-s + 0.584i·6-s + 0.965·7-s − 0.986i·8-s + 0.333·9-s + 0.664i·10-s − 0.211i·11-s + 0.0147·12-s − 0.624i·13-s − 0.977i·14-s + 0.378·15-s − 1.02·16-s + 1.52·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.989 - 0.142i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ -0.989 - 0.142i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.093068079\)
\(L(\frac12)\) \(\approx\) \(1.093068079\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19T \)
59 \( 1 + (3.44e3 + 496. i)T \)
good2 \( 1 + 4.05iT - 16T^{2} \)
5 \( 1 + 16.4T + 625T^{2} \)
7 \( 1 - 47.3T + 2.40e3T^{2} \)
11 \( 1 + 25.6iT - 1.46e4T^{2} \)
13 \( 1 + 105. iT - 2.85e4T^{2} \)
17 \( 1 - 441.T + 8.35e4T^{2} \)
19 \( 1 + 560.T + 1.30e5T^{2} \)
23 \( 1 + 764. iT - 2.79e5T^{2} \)
29 \( 1 + 1.38e3T + 7.07e5T^{2} \)
31 \( 1 - 950. iT - 9.23e5T^{2} \)
37 \( 1 + 632. iT - 1.87e6T^{2} \)
41 \( 1 - 1.02e3T + 2.82e6T^{2} \)
43 \( 1 - 2.95e3iT - 3.41e6T^{2} \)
47 \( 1 + 4.32e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.11e3T + 7.89e6T^{2} \)
61 \( 1 + 4.27e3iT - 1.38e7T^{2} \)
67 \( 1 - 867. iT - 2.01e7T^{2} \)
71 \( 1 + 236.T + 2.54e7T^{2} \)
73 \( 1 - 6.21e3iT - 2.83e7T^{2} \)
79 \( 1 - 8.91e3T + 3.89e7T^{2} \)
83 \( 1 + 1.08e4iT - 4.74e7T^{2} \)
89 \( 1 + 6.26e3iT - 6.27e7T^{2} \)
97 \( 1 + 7.89e3iT - 8.85e7T^{2} \)
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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

−11.34129634624737799231243997808, −10.84508951815289901159108150291, −9.951445205183199197191885828715, −8.392883205671525682111602006555, −7.42903286639907644257245079934, −6.04599066410695034494593276879, −4.63223307143711017873795913018, −3.45664877496578520931764973150, −1.83642194599238158442552633239, −0.42672401791842696483040857210, 1.78538111434115335597177332395, 4.04832678505045336065137196356, 5.26865439749037792450966131092, 6.19299763383188014338367716955, 7.57347769336835160841749279773, 7.85866170235033092477989596110, 9.321824767081502069574722237309, 10.85797496854285970601677540063, 11.49693088725257956274285029812, 12.29985973539582861282746441051

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