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Properties

Label 2-177-177.176-c3-0-35
Degree $2$
Conductor $177$
Sign $-0.645 + 0.763i$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.52·2-s + (0.100 − 5.19i)3-s − 1.60·4-s + 17.9i·5-s + (−0.254 + 13.1i)6-s − 4.91·7-s + 24.2·8-s + (−26.9 − 1.04i)9-s − 45.3i·10-s + 38.0·11-s + (−0.161 + 8.33i)12-s − 51.2i·13-s + 12.4·14-s + (93.1 + 1.80i)15-s − 48.5·16-s − 30.7i·17-s + ⋯
L(s)  = 1  − 0.894·2-s + (0.0193 − 0.999i)3-s − 0.200·4-s + 1.60i·5-s + (−0.0172 + 0.893i)6-s − 0.265·7-s + 1.07·8-s + (−0.999 − 0.0386i)9-s − 1.43i·10-s + 1.04·11-s + (−0.00387 + 0.200i)12-s − 1.09i·13-s + 0.237·14-s + (1.60 + 0.0310i)15-s − 0.759·16-s − 0.438i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.645 + 0.763i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -0.645 + 0.763i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.190188 - 0.410015i\)
\(L(\frac12)\) \(\approx\) \(0.190188 - 0.410015i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.100 + 5.19i)T \)
59 \( 1 + (351. + 285. i)T \)
good2 \( 1 + 2.52T + 8T^{2} \)
5 \( 1 - 17.9iT - 125T^{2} \)
7 \( 1 + 4.91T + 343T^{2} \)
11 \( 1 - 38.0T + 1.33e3T^{2} \)
13 \( 1 + 51.2iT - 2.19e3T^{2} \)
17 \( 1 + 30.7iT - 4.91e3T^{2} \)
19 \( 1 + 45.5T + 6.85e3T^{2} \)
23 \( 1 + 137.T + 1.21e4T^{2} \)
29 \( 1 + 12.5iT - 2.43e4T^{2} \)
31 \( 1 + 75.5iT - 2.97e4T^{2} \)
37 \( 1 + 152. iT - 5.06e4T^{2} \)
41 \( 1 + 272. iT - 6.89e4T^{2} \)
43 \( 1 + 364. iT - 7.95e4T^{2} \)
47 \( 1 - 430.T + 1.03e5T^{2} \)
53 \( 1 + 660. iT - 1.48e5T^{2} \)
61 \( 1 - 251. iT - 2.26e5T^{2} \)
67 \( 1 + 442. iT - 3.00e5T^{2} \)
71 \( 1 - 639. iT - 3.57e5T^{2} \)
73 \( 1 + 36.1iT - 3.89e5T^{2} \)
79 \( 1 + 485.T + 4.93e5T^{2} \)
83 \( 1 + 1.37e3T + 5.71e5T^{2} \)
89 \( 1 - 1.25e3T + 7.04e5T^{2} \)
97 \( 1 + 1.68e3iT - 9.12e5T^{2} \)
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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

−11.69947831012838124699107841239, −10.73750504709877341562979985756, −9.918195698762134131263134382678, −8.711067000969785721528793266059, −7.65466373894902075554380908265, −6.92708094474047034616646146379, −5.89202384376983105838471960976, −3.63882063871827134009968272026, −2.14863090890671546774096549016, −0.29739590678801375353147446739, 1.40604776929211838155208429183, 4.11925382196543540383251725931, 4.63849177639887206519215538607, 6.15211807289396148746209276799, 8.045730439820298635160955397285, 8.915333120973526406251240438552, 9.308187399075556773687109122865, 10.18360893010793037038541635120, 11.47910512600411192287210316848, 12.43060568660747599237265784094

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