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Properties

Label 2-177-59.58-c10-0-80
Degree $2$
Conductor $177$
Sign $-0.950 - 0.309i$
Analytic cond. $112.458$
Root an. cond. $10.6046$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 55.7i·2-s + 140.·3-s − 2.08e3·4-s + 5.22e3·5-s − 7.81e3i·6-s − 1.72e3·7-s + 5.89e4i·8-s + 1.96e4·9-s − 2.91e5i·10-s + 1.56e5i·11-s − 2.91e5·12-s − 3.73e5i·13-s + 9.58e4i·14-s + 7.32e5·15-s + 1.15e6·16-s − 2.44e6·17-s + ⋯
L(s)  = 1  − 1.74i·2-s + 0.577·3-s − 2.03·4-s + 1.67·5-s − 1.00i·6-s − 0.102·7-s + 1.79i·8-s + 0.333·9-s − 2.91i·10-s + 0.969i·11-s − 1.17·12-s − 1.00i·13-s + 0.178i·14-s + 0.964·15-s + 1.09·16-s − 1.72·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.309i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.950 - 0.309i$
Analytic conductor: \(112.458\)
Root analytic conductor: \(10.6046\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5),\ -0.950 - 0.309i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.854032756\)
\(L(\frac12)\) \(\approx\) \(2.854032756\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 140.T \)
59 \( 1 + (6.79e8 + 2.21e8i)T \)
good2 \( 1 + 55.7iT - 1.02e3T^{2} \)
5 \( 1 - 5.22e3T + 9.76e6T^{2} \)
7 \( 1 + 1.72e3T + 2.82e8T^{2} \)
11 \( 1 - 1.56e5iT - 2.59e10T^{2} \)
13 \( 1 + 3.73e5iT - 1.37e11T^{2} \)
17 \( 1 + 2.44e6T + 2.01e12T^{2} \)
19 \( 1 - 3.77e6T + 6.13e12T^{2} \)
23 \( 1 - 6.63e5iT - 4.14e13T^{2} \)
29 \( 1 + 2.35e7T + 4.20e14T^{2} \)
31 \( 1 + 7.44e6iT - 8.19e14T^{2} \)
37 \( 1 + 1.27e8iT - 4.80e15T^{2} \)
41 \( 1 - 1.98e8T + 1.34e16T^{2} \)
43 \( 1 + 2.76e8iT - 2.16e16T^{2} \)
47 \( 1 - 7.61e7iT - 5.25e16T^{2} \)
53 \( 1 - 4.15e8T + 1.74e17T^{2} \)
61 \( 1 + 4.25e8iT - 7.13e17T^{2} \)
67 \( 1 + 2.68e9iT - 1.82e18T^{2} \)
71 \( 1 + 1.83e9T + 3.25e18T^{2} \)
73 \( 1 + 1.40e9iT - 4.29e18T^{2} \)
79 \( 1 - 9.59e8T + 9.46e18T^{2} \)
83 \( 1 + 6.82e9iT - 1.55e19T^{2} \)
89 \( 1 + 6.05e9iT - 3.11e19T^{2} \)
97 \( 1 + 2.41e9iT - 7.37e19T^{2} \)
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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

−10.29571485084665555885790952426, −9.359509087814818586676393262456, −9.147168529743995420208533885729, −7.36643239743511368432922256601, −5.74441903472254000508825114892, −4.64682182682886825222293047903, −3.32709764068071321991011562700, −2.23089366425463282061899451581, −1.85865870810042605283014521637, −0.52400242252978567281546919283, 1.29574415195402176108549346996, 2.69438229420239023457662967985, 4.39374890319468276911289619268, 5.50657710797989603973817777870, 6.31527552585838115523188052678, 7.05133717831419955598849514295, 8.364779762792767685083558647595, 9.226899114966045085840974492832, 9.644381540363534233039103661890, 11.22808447896392847376324217819

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