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Properties

Label 2-90-5.4-c15-0-6
Degree $2$
Conductor $90$
Sign $-0.812 - 0.583i$
Analytic cond. $128.424$
Root an. cond. $11.3324$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 128i·2-s − 1.63e4·4-s + (1.01e5 − 1.41e5i)5-s − 8.59e5i·7-s − 2.09e6i·8-s + (1.81e7 + 1.30e7i)10-s − 2.70e7·11-s + 3.33e8i·13-s + 1.10e8·14-s + 2.68e8·16-s − 1.06e9i·17-s − 2.29e9·19-s + (−1.66e9 + 2.32e9i)20-s − 3.46e9i·22-s − 1.25e10i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.583 − 0.812i)5-s − 0.394i·7-s − 0.353i·8-s + (0.574 + 0.412i)10-s − 0.418·11-s + 1.47i·13-s + 0.278·14-s + 0.250·16-s − 0.631i·17-s − 0.589·19-s + (−0.291 + 0.406i)20-s − 0.295i·22-s − 0.765i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.812 - 0.583i$
Analytic conductor: \(128.424\)
Root analytic conductor: \(11.3324\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :15/2),\ -0.812 - 0.583i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.8903910456\)
\(L(\frac12)\) \(\approx\) \(0.8903910456\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 128iT \)
3 \( 1 \)
5 \( 1 + (-1.01e5 + 1.41e5i)T \)
good7 \( 1 + 8.59e5iT - 4.74e12T^{2} \)
11 \( 1 + 2.70e7T + 4.17e15T^{2} \)
13 \( 1 - 3.33e8iT - 5.11e16T^{2} \)
17 \( 1 + 1.06e9iT - 2.86e18T^{2} \)
19 \( 1 + 2.29e9T + 1.51e19T^{2} \)
23 \( 1 + 1.25e10iT - 2.66e20T^{2} \)
29 \( 1 + 5.43e10T + 8.62e21T^{2} \)
31 \( 1 - 1.05e11T + 2.34e22T^{2} \)
37 \( 1 + 1.33e11iT - 3.33e23T^{2} \)
41 \( 1 + 1.24e12T + 1.55e24T^{2} \)
43 \( 1 - 4.58e11iT - 3.17e24T^{2} \)
47 \( 1 - 2.31e12iT - 1.20e25T^{2} \)
53 \( 1 - 1.66e12iT - 7.31e25T^{2} \)
59 \( 1 - 2.60e13T + 3.65e26T^{2} \)
61 \( 1 + 1.95e13T + 6.02e26T^{2} \)
67 \( 1 - 2.32e13iT - 2.46e27T^{2} \)
71 \( 1 + 8.64e13T + 5.87e27T^{2} \)
73 \( 1 - 1.28e14iT - 8.90e27T^{2} \)
79 \( 1 + 1.07e14T + 2.91e28T^{2} \)
83 \( 1 - 2.06e14iT - 6.11e28T^{2} \)
89 \( 1 - 3.44e14T + 1.74e29T^{2} \)
97 \( 1 - 5.81e14iT - 6.33e29T^{2} \)
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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

−11.72682362833885157092539801846, −10.28125494550851704649218142362, −9.261573618812143779772117735024, −8.445720455053386426027419066136, −7.14835872645969514463430024024, −6.15657656609918794772455812234, −4.94701281390599463984799822313, −4.13775550232952402740590574849, −2.29260432882821868512934212266, −1.02682468564450770056846822464, 0.19080028083976790169923159494, 1.66859742654373951101698572003, 2.67443211349921913524186772870, 3.56764578414355501053123859307, 5.21825888946724132092080246306, 6.11433931316782268937642463379, 7.60133497948923725930920089837, 8.748839598473894145110871166352, 10.08577136542627922568130755892, 10.55970723932533853232717058641

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