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Properties

Label 2-90-5.4-c11-0-7
Degree $2$
Conductor $90$
Sign $-0.285 - 0.958i$
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 + (−6.69e3 + 1.99e3i)5-s + 1.00e4i·7-s − 3.27e4i·8-s + (−6.37e4 − 2.14e5i)10-s + 1.33e5·11-s + 1.52e5i·13-s − 3.20e5·14-s + 1.04e6·16-s − 8.14e6i·17-s + 9.95e6·19-s + (6.85e6 − 2.04e6i)20-s + 4.26e6i·22-s − 1.88e7i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.958 + 0.285i)5-s + 0.224i·7-s − 0.353i·8-s + (−0.201 − 0.677i)10-s + 0.249·11-s + 0.114i·13-s − 0.159·14-s + 0.250·16-s − 1.39i·17-s + 0.922·19-s + (0.479 − 0.142i)20-s + 0.176i·22-s − 0.611i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.285 - 0.958i$
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: \((2,\ 90,\ (\ :11/2),\ -0.285 - 0.958i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.770528 + 1.03314i\)
\(L(\frac12)\) \(\approx\) \(0.770528 + 1.03314i\)
\(L(\frac{13}{2})\) 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 \( 1 + (6.69e3 - 1.99e3i)T \)
good7 \( 1 - 1.00e4iT - 1.97e9T^{2} \)
11 \( 1 - 1.33e5T + 2.85e11T^{2} \)
13 \( 1 - 1.52e5iT - 1.79e12T^{2} \)
17 \( 1 + 8.14e6iT - 3.42e13T^{2} \)
19 \( 1 - 9.95e6T + 1.16e14T^{2} \)
23 \( 1 + 1.88e7iT - 9.52e14T^{2} \)
29 \( 1 + 1.92e8T + 1.22e16T^{2} \)
31 \( 1 - 1.47e8T + 2.54e16T^{2} \)
37 \( 1 - 3.96e8iT - 1.77e17T^{2} \)
41 \( 1 + 1.20e8T + 5.50e17T^{2} \)
43 \( 1 - 7.63e8iT - 9.29e17T^{2} \)
47 \( 1 + 1.46e9iT - 2.47e18T^{2} \)
53 \( 1 - 3.22e9iT - 9.26e18T^{2} \)
59 \( 1 - 8.47e9T + 3.01e19T^{2} \)
61 \( 1 + 4.24e9T + 4.35e19T^{2} \)
67 \( 1 - 1.37e10iT - 1.22e20T^{2} \)
71 \( 1 - 1.36e9T + 2.31e20T^{2} \)
73 \( 1 - 6.85e9iT - 3.13e20T^{2} \)
79 \( 1 + 1.88e10T + 7.47e20T^{2} \)
83 \( 1 - 4.90e10iT - 1.28e21T^{2} \)
89 \( 1 + 5.69e10T + 2.77e21T^{2} \)
97 \( 1 + 2.16e10iT - 7.15e21T^{2} \)
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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

−12.08488661988599326113045959807, −11.33083060571293400638646187945, −9.840945272622960399094635608536, −8.726742486357766061479703558134, −7.60304855813482121258549158052, −6.79615841853913954573628808838, −5.34531447592861319930044728017, −4.15513896986277300227618985086, −2.86192451111426171538836821339, −0.805623940563430256714706042358, 0.45728302825354812072664106244, 1.67186129840013679256169941068, 3.37899960515591531786752491120, 4.18279950242912583213747555373, 5.59025241954532201822858780990, 7.31310118848058413550251874207, 8.331772632313763074938968156538, 9.452763597708544577404452865322, 10.69596296443772508531567428013, 11.57357611854949980282063021134

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