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Properties

Label 2-90-5.4-c11-0-4
Degree $2$
Conductor $90$
Sign $0.995 - 0.0945i$
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 + (−660. − 6.95e3i)5-s + 4.32e4i·7-s + 3.27e4i·8-s + (−2.22e5 + 2.11e4i)10-s − 5.97e5·11-s − 2.59e6i·13-s + 1.38e6·14-s + 1.04e6·16-s + 4.95e6i·17-s − 2.72e6·19-s + (6.76e5 + 7.12e6i)20-s + 1.91e7i·22-s + 3.08e7i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.0945 − 0.995i)5-s + 0.973i·7-s + 0.353i·8-s + (−0.703 + 0.0668i)10-s − 1.11·11-s − 1.94i·13-s + 0.688·14-s + 0.250·16-s + 0.846i·17-s − 0.252·19-s + (0.0472 + 0.497i)20-s + 0.791i·22-s + 1.00i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0945i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.995 - 0.0945i$
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: \((2,\ 90,\ (\ :11/2),\ 0.995 - 0.0945i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.09440 + 0.0518481i\)
\(L(\frac12)\) \(\approx\) \(1.09440 + 0.0518481i\)
\(L(\frac{13}{2})\) 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 \( 1 + (660. + 6.95e3i)T \)
good7 \( 1 - 4.32e4iT - 1.97e9T^{2} \)
11 \( 1 + 5.97e5T + 2.85e11T^{2} \)
13 \( 1 + 2.59e6iT - 1.79e12T^{2} \)
17 \( 1 - 4.95e6iT - 3.42e13T^{2} \)
19 \( 1 + 2.72e6T + 1.16e14T^{2} \)
23 \( 1 - 3.08e7iT - 9.52e14T^{2} \)
29 \( 1 - 9.20e7T + 1.22e16T^{2} \)
31 \( 1 + 6.27e7T + 2.54e16T^{2} \)
37 \( 1 - 2.06e8iT - 1.77e17T^{2} \)
41 \( 1 + 1.18e9T + 5.50e17T^{2} \)
43 \( 1 + 1.09e9iT - 9.29e17T^{2} \)
47 \( 1 - 1.88e9iT - 2.47e18T^{2} \)
53 \( 1 - 3.89e9iT - 9.26e18T^{2} \)
59 \( 1 - 4.79e9T + 3.01e19T^{2} \)
61 \( 1 - 9.24e9T + 4.35e19T^{2} \)
67 \( 1 + 1.72e10iT - 1.22e20T^{2} \)
71 \( 1 - 2.12e10T + 2.31e20T^{2} \)
73 \( 1 - 2.83e10iT - 3.13e20T^{2} \)
79 \( 1 - 5.23e10T + 7.47e20T^{2} \)
83 \( 1 + 4.18e9iT - 1.28e21T^{2} \)
89 \( 1 - 2.02e10T + 2.77e21T^{2} \)
97 \( 1 + 2.95e10iT - 7.15e21T^{2} \)
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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

−12.15813910364143772440976580863, −10.78961456002319867217298589493, −9.850761494830415230893310061055, −8.557411931146690196727733312458, −7.964216072753435682020733955339, −5.67684281706109359774561954499, −5.07621939638853667971693448803, −3.42579412310642919560787676700, −2.22532792897670489848316242597, −0.842974420552451680158576322602, 0.34353330982910150074177840531, 2.26317848596612693313336196138, 3.78420079345476970694651441218, 4.92086713375326252290408508317, 6.62961905344365030428285996395, 7.08898892372401345768736766764, 8.317379350136148566156845186483, 9.756303381089238303035605258190, 10.66918345676618283313552970417, 11.75127978332046241914214638878

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