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Properties

Label 2-90-5.4-c3-0-1
Degree $2$
Conductor $90$
Sign $-0.983 - 0.178i$
Analytic cond. $5.31017$
Root an. cond. $2.30438$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 4·4-s + (−2 + 11i)5-s + 2i·7-s − 8i·8-s + (−22 − 4i)10-s − 70·11-s + 54i·13-s − 4·14-s + 16·16-s − 22i·17-s − 24·19-s + (8 − 44i)20-s − 140i·22-s + 100i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.178 + 0.983i)5-s + 0.107i·7-s − 0.353i·8-s + (−0.695 − 0.126i)10-s − 1.91·11-s + 1.15i·13-s − 0.0763·14-s + 0.250·16-s − 0.313i·17-s − 0.289·19-s + (0.0894 − 0.491i)20-s − 1.35i·22-s + 0.906i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.983 - 0.178i$
Analytic conductor: \(5.31017\)
Root analytic conductor: \(2.30438\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :3/2),\ -0.983 - 0.178i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0798739 + 0.885815i\)
\(L(\frac12)\) \(\approx\) \(0.0798739 + 0.885815i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 \)
5 \( 1 + (2 - 11i)T \)
good7 \( 1 - 2iT - 343T^{2} \)
11 \( 1 + 70T + 1.33e3T^{2} \)
13 \( 1 - 54iT - 2.19e3T^{2} \)
17 \( 1 + 22iT - 4.91e3T^{2} \)
19 \( 1 + 24T + 6.85e3T^{2} \)
23 \( 1 - 100iT - 1.21e4T^{2} \)
29 \( 1 - 216T + 2.43e4T^{2} \)
31 \( 1 - 208T + 2.97e4T^{2} \)
37 \( 1 - 254iT - 5.06e4T^{2} \)
41 \( 1 - 206T + 6.89e4T^{2} \)
43 \( 1 - 292iT - 7.95e4T^{2} \)
47 \( 1 + 320iT - 1.03e5T^{2} \)
53 \( 1 - 402iT - 1.48e5T^{2} \)
59 \( 1 + 370T + 2.05e5T^{2} \)
61 \( 1 + 550T + 2.26e5T^{2} \)
67 \( 1 + 728iT - 3.00e5T^{2} \)
71 \( 1 - 540T + 3.57e5T^{2} \)
73 \( 1 - 604iT - 3.89e5T^{2} \)
79 \( 1 + 792T + 4.93e5T^{2} \)
83 \( 1 + 404iT - 5.71e5T^{2} \)
89 \( 1 + 938T + 7.04e5T^{2} \)
97 \( 1 + 56iT - 9.12e5T^{2} \)
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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

−14.10781147973330493133395981977, −13.44013595304228410489311100526, −11.99168109912117979018073388800, −10.76332721886548661014504211752, −9.778562641608067821768298416545, −8.250181407692054391244660353010, −7.28779139547374256604122749654, −6.14531695116063254432991272069, −4.67076659534695145022232315746, −2.79227728644389770554895895844, 0.51702683540305336395139394427, 2.68573485585703268549144005533, 4.52716288594949031469371517418, 5.61540191644989400971545842340, 7.83827582872334273059172737378, 8.578051630475317594238774977570, 10.10642314008639750098482114518, 10.78724315461149715897162670577, 12.35502594416933827554039524549, 12.83794671659031800713798951246

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