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Properties

Label 2-177-59.58-c8-0-75
Degree $2$
Conductor $177$
Sign $-0.345 - 0.938i$
Analytic cond. $72.1060$
Root an. cond. $8.49152$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 20.7i·2-s + 46.7·3-s − 176.·4-s − 602.·5-s − 972. i·6-s + 1.28e3·7-s − 1.65e3i·8-s + 2.18e3·9-s + 1.25e4i·10-s − 1.94e4i·11-s − 8.24e3·12-s + 3.14e4i·13-s − 2.66e4i·14-s − 2.81e4·15-s − 7.95e4·16-s + 1.37e5·17-s + ⋯
L(s)  = 1  − 1.29i·2-s + 0.577·3-s − 0.688·4-s − 0.963·5-s − 0.750i·6-s + 0.533·7-s − 0.404i·8-s + 0.333·9-s + 1.25i·10-s − 1.32i·11-s − 0.397·12-s + 1.10i·13-s − 0.693i·14-s − 0.556·15-s − 1.21·16-s + 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.345 - 0.938i$
Analytic conductor: \(72.1060\)
Root analytic conductor: \(8.49152\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :4),\ -0.345 - 0.938i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.052840247\)
\(L(\frac12)\) \(\approx\) \(1.052840247\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 46.7T \)
59 \( 1 + (-4.18e6 - 1.13e7i)T \)
good2 \( 1 + 20.7iT - 256T^{2} \)
5 \( 1 + 602.T + 3.90e5T^{2} \)
7 \( 1 - 1.28e3T + 5.76e6T^{2} \)
11 \( 1 + 1.94e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.14e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.37e5T + 6.97e9T^{2} \)
19 \( 1 + 1.49e5T + 1.69e10T^{2} \)
23 \( 1 + 4.11e5iT - 7.83e10T^{2} \)
29 \( 1 + 2.88e5T + 5.00e11T^{2} \)
31 \( 1 + 1.63e6iT - 8.52e11T^{2} \)
37 \( 1 - 2.14e6iT - 3.51e12T^{2} \)
41 \( 1 - 9.60e5T + 7.98e12T^{2} \)
43 \( 1 - 2.15e4iT - 1.16e13T^{2} \)
47 \( 1 + 1.06e5iT - 2.38e13T^{2} \)
53 \( 1 + 1.48e7T + 6.22e13T^{2} \)
61 \( 1 + 1.74e6iT - 1.91e14T^{2} \)
67 \( 1 - 5.44e6iT - 4.06e14T^{2} \)
71 \( 1 + 2.69e7T + 6.45e14T^{2} \)
73 \( 1 + 2.23e7iT - 8.06e14T^{2} \)
79 \( 1 + 1.27e7T + 1.51e15T^{2} \)
83 \( 1 - 6.43e7iT - 2.25e15T^{2} \)
89 \( 1 - 3.11e7iT - 3.93e15T^{2} \)
97 \( 1 + 3.65e7iT - 7.83e15T^{2} \)
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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

−10.78807734778656273398373856523, −9.667882649312131773287714433489, −8.550159283480031540013707503018, −7.78931345012707229028323976195, −6.33012965453291858710444030728, −4.42547888355232792935458576654, −3.67745911788915595214313459889, −2.62659095745269040787285156002, −1.38647271434051337410621671899, −0.22645883971887595278800671301, 1.69325733925756966079176009558, 3.34511670781468464172959440203, 4.62752000200928331843966737848, 5.62487518473272257501440245972, 7.15079240585875891370717523513, 7.73163282018622370618943672836, 8.300386446320461739353768890135, 9.609349128236546168229057182862, 10.83546584853756634943647616952, 12.05926879077096390837100787012

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