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Properties

Label 2-177-177.176-c5-0-27
Degree $2$
Conductor $177$
Sign $0.983 + 0.183i$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.86 − 15.3i)3-s − 32·4-s + 49.9i·5-s − 258.·7-s + (−226. − 87.6i)9-s + (−91.5 + 490. i)12-s + (764. + 142. i)15-s + 1.02e3·16-s − 683. i·17-s + 896.·19-s − 1.59e3i·20-s + (−739. + 3.96e3i)21-s + 633.·25-s + (−1.99e3 + 3.22e3i)27-s + 8.27e3·28-s + 3.24e3i·29-s + ⋯
L(s)  = 1  + (0.183 − 0.983i)3-s − 4-s + 0.892i·5-s − 1.99·7-s + (−0.932 − 0.360i)9-s + (−0.183 + 0.983i)12-s + (0.877 + 0.163i)15-s + 16-s − 0.573i·17-s + 0.569·19-s − 0.892i·20-s + (−0.366 + 1.96i)21-s + 0.202·25-s + (−0.525 + 0.850i)27-s + 1.99·28-s + 0.716i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.983 + 0.183i$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 0.983 + 0.183i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8339405406\)
\(L(\frac12)\) \(\approx\) \(0.8339405406\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.86 + 15.3i)T \)
59 \( 1 + 2.67e4iT \)
good2 \( 1 + 32T^{2} \)
5 \( 1 - 49.9iT - 3.12e3T^{2} \)
7 \( 1 + 258.T + 1.68e4T^{2} \)
11 \( 1 + 1.61e5T^{2} \)
13 \( 1 - 3.71e5T^{2} \)
17 \( 1 + 683. iT - 1.41e6T^{2} \)
19 \( 1 - 896.T + 2.47e6T^{2} \)
23 \( 1 + 6.43e6T^{2} \)
29 \( 1 - 3.24e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.86e7T^{2} \)
37 \( 1 - 6.93e7T^{2} \)
41 \( 1 + 1.89e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.47e8T^{2} \)
47 \( 1 + 2.29e8T^{2} \)
53 \( 1 - 3.88e4iT - 4.18e8T^{2} \)
61 \( 1 - 8.44e8T^{2} \)
67 \( 1 - 1.35e9T^{2} \)
71 \( 1 - 5.94e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.07e9T^{2} \)
79 \( 1 - 9.68e4T + 3.07e9T^{2} \)
83 \( 1 + 3.93e9T^{2} \)
89 \( 1 + 5.58e9T^{2} \)
97 \( 1 - 8.58e9T^{2} \)
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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

−12.17091503078063237256315904961, −10.64309598584144918699037118574, −9.594245969108183982232889377730, −8.873348688376130404954503814241, −7.40182431523670109167041251588, −6.65289216354584844539371430711, −5.60940791421029063036682178827, −3.58366243921556658688691226213, −2.79199418591818840342852145919, −0.61780575586255061420920796622, 0.50366992135817344513325016886, 3.13955587279269194280119004764, 4.05747064257536914831777519991, 5.17662213856628252579482979317, 6.25838545534191833109995519271, 8.129824788387795597931209572356, 9.144465433626853279034278621073, 9.604486743201143930796677914141, 10.38524875733519203901941472901, 12.03192249500921121902691688446

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