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Properties

Label 2-90-5.3-c4-0-4
Degree $2$
Conductor $90$
Sign $0.767 + 0.640i$
Analytic cond. $9.30329$
Root an. cond. $3.05013$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 2i)2-s − 8i·4-s + (15 + 20i)5-s + (29 − 29i)7-s + (−16 − 16i)8-s + (70 + 10i)10-s + 118·11-s + (69 + 69i)13-s − 116i·14-s − 64·16-s + (271 − 271i)17-s − 280i·19-s + (160 − 120i)20-s + (236 − 236i)22-s + (−269 − 269i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.5i·4-s + (0.599 + 0.800i)5-s + (0.591 − 0.591i)7-s + (−0.250 − 0.250i)8-s + (0.700 + 0.100i)10-s + 0.975·11-s + (0.408 + 0.408i)13-s − 0.591i·14-s − 0.250·16-s + (0.937 − 0.937i)17-s − 0.775i·19-s + (0.400 − 0.299i)20-s + (0.487 − 0.487i)22-s + (−0.508 − 0.508i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.767 + 0.640i$
Analytic conductor: \(9.30329\)
Root analytic conductor: \(3.05013\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :2),\ 0.767 + 0.640i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.47486 - 0.897050i\)
\(L(\frac12)\) \(\approx\) \(2.47486 - 0.897050i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 2i)T \)
3 \( 1 \)
5 \( 1 + (-15 - 20i)T \)
good7 \( 1 + (-29 + 29i)T - 2.40e3iT^{2} \)
11 \( 1 - 118T + 1.46e4T^{2} \)
13 \( 1 + (-69 - 69i)T + 2.85e4iT^{2} \)
17 \( 1 + (-271 + 271i)T - 8.35e4iT^{2} \)
19 \( 1 + 280iT - 1.30e5T^{2} \)
23 \( 1 + (269 + 269i)T + 2.79e5iT^{2} \)
29 \( 1 - 680iT - 7.07e5T^{2} \)
31 \( 1 - 202T + 9.23e5T^{2} \)
37 \( 1 + (651 - 651i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.68e3T + 2.82e6T^{2} \)
43 \( 1 + (-1.08e3 - 1.08e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (1.26e3 - 1.26e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-611 - 611i)T + 7.89e6iT^{2} \)
59 \( 1 - 1.16e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.59e3T + 1.38e7T^{2} \)
67 \( 1 + (751 - 751i)T - 2.01e7iT^{2} \)
71 \( 1 + 6.44e3T + 2.54e7T^{2} \)
73 \( 1 + (2.95e3 + 2.95e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 1.05e4iT - 3.89e7T^{2} \)
83 \( 1 + (-6.23e3 - 6.23e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 1.44e4iT - 6.27e7T^{2} \)
97 \( 1 + (7.31e3 - 7.31e3i)T - 8.85e7iT^{2} \)
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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

−13.55375733280712403481734816842, −12.03636380871631201522049746104, −11.16816482534336306931737782772, −10.22164564417411389095421472431, −9.100787338737262232997105266530, −7.29569000275293486788477076260, −6.19485264190331227415684128970, −4.65342879064662208317934131367, −3.15776061458109516474282197797, −1.43673412339740751397714452451, 1.64183786469500758518503405493, 3.86638745644093820512550840953, 5.36560852041703398020676639355, 6.16885985235360250074070289607, 7.962290289517303540004830603790, 8.817480844959391405457780138397, 10.10351302859064704004317790970, 11.77503048904604526139005289835, 12.46443309070672284384089088869, 13.62295191399786828515165909406

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