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Properties

Label 2-177-59.58-c10-0-98
Degree $2$
Conductor $177$
Sign $0.858 - 0.512i$
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  − 50.8i·2-s + 140.·3-s − 1.56e3·4-s + 2.10e3·5-s − 7.14e3i·6-s − 2.64e4·7-s + 2.75e4i·8-s + 1.96e4·9-s − 1.07e5i·10-s − 1.41e5i·11-s − 2.19e5·12-s − 2.58e5i·13-s + 1.34e6i·14-s + 2.95e5·15-s − 1.99e5·16-s + 6.00e5·17-s + ⋯
L(s)  = 1  − 1.59i·2-s + 0.577·3-s − 1.52·4-s + 0.674·5-s − 0.918i·6-s − 1.57·7-s + 0.841i·8-s + 0.333·9-s − 1.07i·10-s − 0.878i·11-s − 0.882·12-s − 0.696i·13-s + 2.50i·14-s + 0.389·15-s − 0.190·16-s + 0.422·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.858 - 0.512i$
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: \((2,\ 177,\ (\ :5),\ 0.858 - 0.512i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.1253771608\)
\(L(\frac12)\) \(\approx\) \(0.1253771608\)
\(L(6)\) 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 \( 1 + (-6.14e8 + 3.66e8i)T \)
good2 \( 1 + 50.8iT - 1.02e3T^{2} \)
5 \( 1 - 2.10e3T + 9.76e6T^{2} \)
7 \( 1 + 2.64e4T + 2.82e8T^{2} \)
11 \( 1 + 1.41e5iT - 2.59e10T^{2} \)
13 \( 1 + 2.58e5iT - 1.37e11T^{2} \)
17 \( 1 - 6.00e5T + 2.01e12T^{2} \)
19 \( 1 + 4.30e4T + 6.13e12T^{2} \)
23 \( 1 + 1.84e6iT - 4.14e13T^{2} \)
29 \( 1 - 3.28e6T + 4.20e14T^{2} \)
31 \( 1 + 6.86e6iT - 8.19e14T^{2} \)
37 \( 1 + 3.37e7iT - 4.80e15T^{2} \)
41 \( 1 + 1.94e8T + 1.34e16T^{2} \)
43 \( 1 + 7.75e7iT - 2.16e16T^{2} \)
47 \( 1 - 1.56e8iT - 5.25e16T^{2} \)
53 \( 1 + 1.74e8T + 1.74e17T^{2} \)
61 \( 1 + 1.60e8iT - 7.13e17T^{2} \)
67 \( 1 - 1.29e9iT - 1.82e18T^{2} \)
71 \( 1 + 1.68e9T + 3.25e18T^{2} \)
73 \( 1 - 3.06e9iT - 4.29e18T^{2} \)
79 \( 1 - 1.15e9T + 9.46e18T^{2} \)
83 \( 1 - 7.52e9iT - 1.55e19T^{2} \)
89 \( 1 - 5.42e9iT - 3.11e19T^{2} \)
97 \( 1 + 3.52e9iT - 7.37e19T^{2} \)
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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

−9.951851855089390664033852802804, −9.419503982510695330294637926124, −8.369738357664420126184549374866, −6.71232104947516123198636526648, −5.55587186042866334610974853724, −3.82632486157652328753362175025, −3.13177241957582394628775652660, −2.34865039315976740967543609787, −1.03327065841145481540830601861, −0.02528701627982734300914469214, 1.90924543575618062108673055112, 3.32783294125942406803571859207, 4.65455382410790266184066433089, 5.90750996760060038217485568150, 6.67917357279150632795452372588, 7.40134708114733838585780266915, 8.661387184890621254348717429963, 9.541436530518430816620978886503, 10.05806275906049322215532937091, 12.06268144442187053057936761046

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