Properties

Degree $2$
Conductor $90$
Sign $0.178 - 0.983i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s − 16·4-s + (−55 − 10i)5-s − 158i·7-s + 64i·8-s + (−40 + 220i)10-s + 148·11-s + 684i·13-s − 632·14-s + 256·16-s + 2.04e3i·17-s − 2.22e3·19-s + (880 + 160i)20-s − 592i·22-s + 1.24e3i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.983 − 0.178i)5-s − 1.21i·7-s + 0.353i·8-s + (−0.126 + 0.695i)10-s + 0.368·11-s + 1.12i·13-s − 0.861·14-s + 0.250·16-s + 1.71i·17-s − 1.41·19-s + (0.491 + 0.0894i)20-s − 0.260i·22-s + 0.491i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.178 - 0.983i$
Motivic weight: \(5\)
Character: $\chi_{90} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :5/2),\ 0.178 - 0.983i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.314475 + 0.262453i\)
\(L(\frac12)\) \(\approx\) \(0.314475 + 0.262453i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
3 \( 1 \)
5 \( 1 + (55 + 10i)T \)
good7 \( 1 + 158iT - 1.68e4T^{2} \)
11 \( 1 - 148T + 1.61e5T^{2} \)
13 \( 1 - 684iT - 3.71e5T^{2} \)
17 \( 1 - 2.04e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.22e3T + 2.47e6T^{2} \)
23 \( 1 - 1.24e3iT - 6.43e6T^{2} \)
29 \( 1 + 270T + 2.05e7T^{2} \)
31 \( 1 + 2.04e3T + 2.86e7T^{2} \)
37 \( 1 - 4.37e3iT - 6.93e7T^{2} \)
41 \( 1 - 2.39e3T + 1.15e8T^{2} \)
43 \( 1 - 2.29e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.06e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.96e3iT - 4.18e8T^{2} \)
59 \( 1 + 3.97e4T + 7.14e8T^{2} \)
61 \( 1 + 4.22e4T + 8.44e8T^{2} \)
67 \( 1 + 3.20e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.24e3T + 1.80e9T^{2} \)
73 \( 1 - 3.01e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.52e4T + 3.07e9T^{2} \)
83 \( 1 - 2.78e4iT - 3.93e9T^{2} \)
89 \( 1 + 8.52e4T + 5.58e9T^{2} \)
97 \( 1 - 9.72e4iT - 8.58e9T^{2} \)
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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

−13.18746933666727150660279706217, −12.25527056946384563452647572226, −11.15492254074759359287940705201, −10.44169349266706109599366942044, −8.982643058611419284405275498388, −7.889540861626861394864905455711, −6.57191011569547316844062799138, −4.38166434214523144874307368998, −3.76574800022741387248522764430, −1.51003421998450268117401362950, 0.17294924110467147987588386132, 2.89230975449591630649322238711, 4.58969947940424697057555459734, 5.90046019645149980008757581893, 7.23065331322161692210391381178, 8.333180019243745791419694862899, 9.233753046113321295957484866732, 10.81925603484668640903743472011, 12.01489114720489222406930790558, 12.79815699147473866154743754861

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