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Properties

Label 2-177-3.2-c4-0-38
Degree $2$
Conductor $177$
Sign $0.772 + 0.635i$
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  − 2.69i·2-s + (−5.71 + 6.95i)3-s + 8.75·4-s − 3.65i·5-s + (18.7 + 15.3i)6-s + 22.3·7-s − 66.6i·8-s + (−15.6 − 79.4i)9-s − 9.83·10-s + 122. i·11-s + (−50.0 + 60.8i)12-s − 0.917·13-s − 60.1i·14-s + (25.4 + 20.8i)15-s − 39.2·16-s − 352. i·17-s + ⋯
L(s)  = 1  − 0.672i·2-s + (−0.635 + 0.772i)3-s + 0.547·4-s − 0.146i·5-s + (0.519 + 0.427i)6-s + 0.456·7-s − 1.04i·8-s + (−0.193 − 0.981i)9-s − 0.0983·10-s + 1.01i·11-s + (−0.347 + 0.422i)12-s − 0.00543·13-s − 0.306i·14-s + (0.112 + 0.0928i)15-s − 0.153·16-s − 1.21i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.772 + 0.635i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.772 + 0.635i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.904355020\)
\(L(\frac12)\) \(\approx\) \(1.904355020\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.71 - 6.95i)T \)
59 \( 1 + 453. iT \)
good2 \( 1 + 2.69iT - 16T^{2} \)
5 \( 1 + 3.65iT - 625T^{2} \)
7 \( 1 - 22.3T + 2.40e3T^{2} \)
11 \( 1 - 122. iT - 1.46e4T^{2} \)
13 \( 1 + 0.917T + 2.85e4T^{2} \)
17 \( 1 + 352. iT - 8.35e4T^{2} \)
19 \( 1 - 133.T + 1.30e5T^{2} \)
23 \( 1 - 754. iT - 2.79e5T^{2} \)
29 \( 1 + 1.37e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.45e3T + 9.23e5T^{2} \)
37 \( 1 - 1.86e3T + 1.87e6T^{2} \)
41 \( 1 + 1.11e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.27e3T + 3.41e6T^{2} \)
47 \( 1 + 1.07e3iT - 4.87e6T^{2} \)
53 \( 1 - 990. iT - 7.89e6T^{2} \)
61 \( 1 - 3.40e3T + 1.38e7T^{2} \)
67 \( 1 + 5.36e3T + 2.01e7T^{2} \)
71 \( 1 - 104. iT - 2.54e7T^{2} \)
73 \( 1 - 1.01e4T + 2.83e7T^{2} \)
79 \( 1 - 4.02e3T + 3.89e7T^{2} \)
83 \( 1 - 1.34e4iT - 4.74e7T^{2} \)
89 \( 1 + 1.16e3iT - 6.27e7T^{2} \)
97 \( 1 + 3.66e3T + 8.85e7T^{2} \)
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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

−11.69669040103310765884638384842, −11.11571985344068265724550240836, −9.934999795544874485758207740273, −9.462978398496257699849376976207, −7.67870938429174780096467509518, −6.56168101648912006598979985894, −5.20134947763217365718785300947, −4.12233657998400373780903666951, −2.63591484024925528901887091102, −0.939916754016605911816685634211, 1.17372857967535645413199926098, 2.72690302283083658820479827717, 4.91078221470080304236186955056, 6.09072956950987536185620703107, 6.66932666907859120969928259728, 7.911684569608663972517229494199, 8.525503616136978219896205197014, 10.60712385919919903819149218273, 11.08020719056197586569589353817, 12.09060889861662278146953816126

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