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Properties

Label 2-90-5.4-c11-0-6
Degree $2$
Conductor $90$
Sign $-0.896 - 0.442i$
Analytic cond. $69.1508$
Root an. cond. $8.31570$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 32i·2-s − 1.02e3·4-s + (3.09e3 − 6.26e3i)5-s + 7.34e4i·7-s − 3.27e4i·8-s + (2.00e5 + 9.89e4i)10-s + 7.52e3·11-s + 1.55e6i·13-s − 2.35e6·14-s + 1.04e6·16-s − 7.69e6i·17-s + 1.25e7·19-s + (−3.16e6 + 6.41e6i)20-s + 2.40e5i·22-s + 8.11e4i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.442 − 0.896i)5-s + 1.65i·7-s − 0.353i·8-s + (0.634 + 0.312i)10-s + 0.0140·11-s + 1.15i·13-s − 1.16·14-s + 0.250·16-s − 1.31i·17-s + 1.16·19-s + (−0.221 + 0.448i)20-s + 0.00996i·22-s + 0.00262i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.896 - 0.442i$
Analytic conductor: \(69.1508\)
Root analytic conductor: \(8.31570\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :11/2),\ -0.896 - 0.442i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.353692 + 1.51619i\)
\(L(\frac12)\) \(\approx\) \(0.353692 + 1.51619i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 32iT \)
3 \( 1 \)
5 \( 1 + (-3.09e3 + 6.26e3i)T \)
good7 \( 1 - 7.34e4iT - 1.97e9T^{2} \)
11 \( 1 - 7.52e3T + 2.85e11T^{2} \)
13 \( 1 - 1.55e6iT - 1.79e12T^{2} \)
17 \( 1 + 7.69e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.25e7T + 1.16e14T^{2} \)
23 \( 1 - 8.11e4iT - 9.52e14T^{2} \)
29 \( 1 - 5.08e7T + 1.22e16T^{2} \)
31 \( 1 + 3.12e7T + 2.54e16T^{2} \)
37 \( 1 - 6.52e8iT - 1.77e17T^{2} \)
41 \( 1 + 1.26e7T + 5.50e17T^{2} \)
43 \( 1 - 1.47e8iT - 9.29e17T^{2} \)
47 \( 1 - 2.34e9iT - 2.47e18T^{2} \)
53 \( 1 - 4.42e8iT - 9.26e18T^{2} \)
59 \( 1 + 1.02e10T + 3.01e19T^{2} \)
61 \( 1 - 2.42e8T + 4.35e19T^{2} \)
67 \( 1 - 2.47e9iT - 1.22e20T^{2} \)
71 \( 1 - 2.58e10T + 2.31e20T^{2} \)
73 \( 1 - 1.06e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.10e10T + 7.47e20T^{2} \)
83 \( 1 + 3.03e10iT - 1.28e21T^{2} \)
89 \( 1 + 7.74e10T + 2.77e21T^{2} \)
97 \( 1 - 1.50e11iT - 7.15e21T^{2} \)
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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

−12.29257378860057724982238948844, −11.65358888100193328545260677803, −9.495892471840042313381318678408, −9.188449152777602675649109972821, −8.049198650572765402125497629986, −6.54620822481298551420217217685, −5.46252164990928794153407969349, −4.69470325723120583547677810015, −2.72221087844582702968156555373, −1.30156622254117453675948327078, 0.39180231695304132195431179344, 1.57502553487279892933909321842, 3.10916243239806954711709061568, 3.96882678271287586210726869264, 5.59671859421486819370719582121, 7.01355521875877095908179096154, 8.004973263467416042985314540689, 9.727504792858567478586630654631, 10.50615775375560541078094456433, 11.02130481503092284656429905782

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