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Properties

Label 2-177-59.58-c6-0-17
Degree $2$
Conductor $177$
Sign $-0.756 + 0.654i$
Analytic cond. $40.7195$
Root an. cond. $6.38118$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14.6i·2-s − 15.5·3-s − 149.·4-s − 98.4·5-s + 227. i·6-s − 154.·7-s + 1.25e3i·8-s + 243·9-s + 1.43e3i·10-s + 1.55e3i·11-s + 2.33e3·12-s − 1.34e3i·13-s + 2.25e3i·14-s + 1.53e3·15-s + 8.74e3·16-s + 345.·17-s + ⋯
L(s)  = 1  − 1.82i·2-s − 0.577·3-s − 2.33·4-s − 0.787·5-s + 1.05i·6-s − 0.449·7-s + 2.44i·8-s + 0.333·9-s + 1.43i·10-s + 1.16i·11-s + 1.35·12-s − 0.613i·13-s + 0.822i·14-s + 0.454·15-s + 2.13·16-s + 0.0702·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.756 + 0.654i$
Analytic conductor: \(40.7195\)
Root analytic conductor: \(6.38118\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3),\ -0.756 + 0.654i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.6517038934\)
\(L(\frac12)\) \(\approx\) \(0.6517038934\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 15.5T \)
59 \( 1 + (-1.55e5 + 1.34e5i)T \)
good2 \( 1 + 14.6iT - 64T^{2} \)
5 \( 1 + 98.4T + 1.56e4T^{2} \)
7 \( 1 + 154.T + 1.17e5T^{2} \)
11 \( 1 - 1.55e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.34e3iT - 4.82e6T^{2} \)
17 \( 1 - 345.T + 2.41e7T^{2} \)
19 \( 1 - 647.T + 4.70e7T^{2} \)
23 \( 1 - 8.41e3iT - 1.48e8T^{2} \)
29 \( 1 - 194.T + 5.94e8T^{2} \)
31 \( 1 - 2.40e3iT - 8.87e8T^{2} \)
37 \( 1 - 2.53e4iT - 2.56e9T^{2} \)
41 \( 1 + 7.97e4T + 4.75e9T^{2} \)
43 \( 1 + 2.99e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.60e5iT - 1.07e10T^{2} \)
53 \( 1 - 9.63e4T + 2.21e10T^{2} \)
61 \( 1 - 1.27e5iT - 5.15e10T^{2} \)
67 \( 1 - 4.01e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.08e5T + 1.28e11T^{2} \)
73 \( 1 - 4.20e5iT - 1.51e11T^{2} \)
79 \( 1 - 1.63e5T + 2.43e11T^{2} \)
83 \( 1 + 7.10e5iT - 3.26e11T^{2} \)
89 \( 1 + 4.05e5iT - 4.96e11T^{2} \)
97 \( 1 + 3.66e5iT - 8.32e11T^{2} \)
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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

−11.45869668673565281931368524103, −10.24871970133337546505737774805, −9.810485459847551895520895050751, −8.474930972921945370031038630331, −7.15920726464966830869608924741, −5.32631591631784981097006760233, −4.22994264578593581655999288556, −3.26655103366786057208605527086, −1.81507480032387157446048680625, −0.43566327939383261041872966522, 0.53479385377259668185173887707, 3.65264705249223745803125451935, 4.75648611380422281209261917726, 5.93366263672049499436113882177, 6.64931176493210609697094522810, 7.69791776746965000124370059351, 8.558305986451166124511086992860, 9.579267682176236168549216464703, 10.99421355047543739989260218778, 12.12121925057200951957775661493

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