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Properties

Label 2-177-3.2-c4-0-45
Degree $2$
Conductor $177$
Sign $-0.988 + 0.151i$
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  − 7.85i·2-s + (1.36 + 8.89i)3-s − 45.7·4-s + 33.8i·5-s + (69.9 − 10.7i)6-s + 13.8·7-s + 233. i·8-s + (−77.2 + 24.2i)9-s + 266.·10-s − 192. i·11-s + (−62.4 − 407. i)12-s − 170.·13-s − 108. i·14-s + (−301. + 46.2i)15-s + 1.10e3·16-s − 492. i·17-s + ⋯
L(s)  = 1  − 1.96i·2-s + (0.151 + 0.988i)3-s − 2.85·4-s + 1.35i·5-s + (1.94 − 0.297i)6-s + 0.281·7-s + 3.65i·8-s + (−0.954 + 0.299i)9-s + 2.66·10-s − 1.58i·11-s + (−0.433 − 2.82i)12-s − 1.01·13-s − 0.553i·14-s + (−1.33 + 0.205i)15-s + 4.31·16-s − 1.70i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.988 + 0.151i$
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: \((2,\ 177,\ (\ :2),\ -0.988 + 0.151i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.8350163739\)
\(L(\frac12)\) \(\approx\) \(0.8350163739\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.36 - 8.89i)T \)
59 \( 1 - 453. iT \)
good2 \( 1 + 7.85iT - 16T^{2} \)
5 \( 1 - 33.8iT - 625T^{2} \)
7 \( 1 - 13.8T + 2.40e3T^{2} \)
11 \( 1 + 192. iT - 1.46e4T^{2} \)
13 \( 1 + 170.T + 2.85e4T^{2} \)
17 \( 1 + 492. iT - 8.35e4T^{2} \)
19 \( 1 - 210.T + 1.30e5T^{2} \)
23 \( 1 + 115. iT - 2.79e5T^{2} \)
29 \( 1 + 439. iT - 7.07e5T^{2} \)
31 \( 1 - 175.T + 9.23e5T^{2} \)
37 \( 1 - 1.95e3T + 1.87e6T^{2} \)
41 \( 1 + 1.81e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.52e3T + 3.41e6T^{2} \)
47 \( 1 - 1.69e3iT - 4.87e6T^{2} \)
53 \( 1 + 873. iT - 7.89e6T^{2} \)
61 \( 1 + 4.97e3T + 1.38e7T^{2} \)
67 \( 1 + 4.01e3T + 2.01e7T^{2} \)
71 \( 1 + 233. iT - 2.54e7T^{2} \)
73 \( 1 + 2.07e3T + 2.83e7T^{2} \)
79 \( 1 - 336.T + 3.89e7T^{2} \)
83 \( 1 - 687. iT - 4.74e7T^{2} \)
89 \( 1 + 5.66e3iT - 6.27e7T^{2} \)
97 \( 1 + 3.42e3T + 8.85e7T^{2} \)
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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

−11.25692395479871674735723886205, −10.79270234599031860377770994966, −9.821925910638491398800685204919, −9.153112280175689888337660133927, −7.85094594540858731724828400196, −5.58975142356073203476689032161, −4.40803725165869599733819792068, −3.05442708124439631500415740410, −2.71924753038239334010319278013, −0.33814948929428494559594449764, 1.31357145407972600415591545923, 4.37212041341801471421098101583, 5.20267832230341674615954315834, 6.31112343584514181200154907187, 7.47338768876846057205429166343, 7.993035669018650135629156890103, 8.976232944675231640630729935456, 9.797172653284687329640190833329, 12.32176814364958383133560674674, 12.69632090839584156520949346812

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