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Properties

Label 2-90-5.4-c11-0-25
Degree $2$
Conductor $90$
Sign $-0.161 - 0.986i$
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 + (−6.89e3 + 1.13e3i)5-s − 1.43e4i·7-s + 3.27e4i·8-s + (3.61e4 + 2.20e5i)10-s + 4.93e5·11-s − 9.84e5i·13-s − 4.60e5·14-s + 1.04e6·16-s − 9.98e6i·17-s − 9.97e6·19-s + (7.06e6 − 1.15e6i)20-s − 1.57e7i·22-s − 4.66e6i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.986 + 0.161i)5-s − 0.323i·7-s + 0.353i·8-s + (0.114 + 0.697i)10-s + 0.924·11-s − 0.735i·13-s − 0.228·14-s + 0.250·16-s − 1.70i·17-s − 0.924·19-s + (0.493 − 0.0809i)20-s − 0.653i·22-s − 0.151i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.161 - 0.986i$
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: \((2,\ 90,\ (\ :11/2),\ -0.161 - 0.986i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0254484 + 0.0299624i\)
\(L(\frac12)\) \(\approx\) \(0.0254484 + 0.0299624i\)
\(L(\frac{13}{2})\) 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 \( 1 + (6.89e3 - 1.13e3i)T \)
good7 \( 1 + 1.43e4iT - 1.97e9T^{2} \)
11 \( 1 - 4.93e5T + 2.85e11T^{2} \)
13 \( 1 + 9.84e5iT - 1.79e12T^{2} \)
17 \( 1 + 9.98e6iT - 3.42e13T^{2} \)
19 \( 1 + 9.97e6T + 1.16e14T^{2} \)
23 \( 1 + 4.66e6iT - 9.52e14T^{2} \)
29 \( 1 + 4.31e7T + 1.22e16T^{2} \)
31 \( 1 - 1.06e8T + 2.54e16T^{2} \)
37 \( 1 + 1.33e8iT - 1.77e17T^{2} \)
41 \( 1 + 1.24e9T + 5.50e17T^{2} \)
43 \( 1 - 1.57e9iT - 9.29e17T^{2} \)
47 \( 1 - 3.78e8iT - 2.47e18T^{2} \)
53 \( 1 + 3.54e9iT - 9.26e18T^{2} \)
59 \( 1 + 6.43e9T + 3.01e19T^{2} \)
61 \( 1 - 2.83e8T + 4.35e19T^{2} \)
67 \( 1 - 7.62e9iT - 1.22e20T^{2} \)
71 \( 1 - 1.48e9T + 2.31e20T^{2} \)
73 \( 1 - 1.24e10iT - 3.13e20T^{2} \)
79 \( 1 + 3.51e10T + 7.47e20T^{2} \)
83 \( 1 - 9.64e9iT - 1.28e21T^{2} \)
89 \( 1 - 4.39e10T + 2.77e21T^{2} \)
97 \( 1 - 1.45e11iT - 7.15e21T^{2} \)
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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

−11.28793477421596929821073509185, −10.19021465633993249741159916803, −8.987788651439238295514279842660, −7.85507948571941922111848789957, −6.68211526935159004244188270329, −4.89023682295322260070746943319, −3.83292699551545675211229175626, −2.74578999442649239685578710627, −1.03137214015693425569582349908, −0.01137351396484860531814232716, 1.61131701621379536192229423231, 3.65985756372080727339330061296, 4.49208336823480620704756614498, 6.05842284485136556413074399947, 7.03617392568787113109806401589, 8.331109131228291371971405472413, 8.962839285555219061378790364298, 10.50139103271891345947822743825, 11.78255318681167959178915780169, 12.56457904346978550374613168526

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