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Properties

Label 2-90-5.4-c11-0-23
Degree $2$
Conductor $90$
Sign $-0.973 + 0.228i$
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 + (−1.59e3 − 6.80e3i)5-s − 2.44e4i·7-s + 3.27e4i·8-s + (−2.17e5 + 5.10e4i)10-s + 6.69e5·11-s − 5.79e5i·13-s − 7.83e5·14-s + 1.04e6·16-s − 7.85e6i·17-s + 1.81e7·19-s + (1.63e6 + 6.96e6i)20-s − 2.14e7i·22-s − 1.97e7i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.228 − 0.973i)5-s − 0.550i·7-s + 0.353i·8-s + (−0.688 + 0.161i)10-s + 1.25·11-s − 0.432i·13-s − 0.389·14-s + 0.250·16-s − 1.34i·17-s + 1.67·19-s + (0.114 + 0.486i)20-s − 0.885i·22-s − 0.638i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.973 + 0.228i$
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: \((2,\ 90,\ (\ :11/2),\ -0.973 + 0.228i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.238082 - 2.05905i\)
\(L(\frac12)\) \(\approx\) \(0.238082 - 2.05905i\)
\(L(\frac{13}{2})\) 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 \( 1 + (1.59e3 + 6.80e3i)T \)
good7 \( 1 + 2.44e4iT - 1.97e9T^{2} \)
11 \( 1 - 6.69e5T + 2.85e11T^{2} \)
13 \( 1 + 5.79e5iT - 1.79e12T^{2} \)
17 \( 1 + 7.85e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.81e7T + 1.16e14T^{2} \)
23 \( 1 + 1.97e7iT - 9.52e14T^{2} \)
29 \( 1 - 2.02e8T + 1.22e16T^{2} \)
31 \( 1 - 1.02e8T + 2.54e16T^{2} \)
37 \( 1 - 5.02e8iT - 1.77e17T^{2} \)
41 \( 1 + 3.44e8T + 5.50e17T^{2} \)
43 \( 1 + 1.43e9iT - 9.29e17T^{2} \)
47 \( 1 + 1.38e9iT - 2.47e18T^{2} \)
53 \( 1 + 2.89e8iT - 9.26e18T^{2} \)
59 \( 1 + 2.47e9T + 3.01e19T^{2} \)
61 \( 1 - 9.16e7T + 4.35e19T^{2} \)
67 \( 1 + 3.09e7iT - 1.22e20T^{2} \)
71 \( 1 + 2.42e10T + 2.31e20T^{2} \)
73 \( 1 - 3.55e9iT - 3.13e20T^{2} \)
79 \( 1 - 1.16e10T + 7.47e20T^{2} \)
83 \( 1 - 2.33e10iT - 1.28e21T^{2} \)
89 \( 1 + 1.02e11T + 2.77e21T^{2} \)
97 \( 1 - 5.87e10iT - 7.15e21T^{2} \)
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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

−11.82216038295906058808985996961, −10.22943928427686655002072015915, −9.320384723000656002471801834587, −8.312150655316179435050311224582, −6.95710423502106703602905382123, −5.22937840549314275375068484350, −4.26079090623659775648624184034, −3.01722989907651524256087263202, −1.22201978876642939518905347834, −0.61506349032387871736958840334, 1.32842556457810966832493401699, 3.06255668592363662493299044802, 4.24430626936263991067189235473, 5.87556113447697916601821059412, 6.68752964223188633071702897223, 7.80133718072661476981113408208, 9.013281898285479374999943214424, 10.05632504261807174563710229068, 11.42881877625928079337372479279, 12.26094217219121893424604705210

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