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Properties

Label 2-90-5.2-c4-0-6
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: $\chi_{90} (37, \cdot )$
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

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

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