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Properties

Label 2-90-15.2-c11-0-21
Degree $2$
Conductor $90$
Sign $-0.644 - 0.764i$
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  + (22.6 − 22.6i)2-s − 1.02e3i·4-s + (−2.17e3 − 6.63e3i)5-s + (−3.54e4 − 3.54e4i)7-s + (−2.31e4 − 2.31e4i)8-s + (−1.99e5 − 1.00e5i)10-s − 9.32e5i·11-s + (8.10e5 − 8.10e5i)13-s − 1.60e6·14-s − 1.04e6·16-s + (2.79e6 − 2.79e6i)17-s − 3.87e6i·19-s + (−6.79e6 + 2.23e6i)20-s + (−2.10e7 − 2.10e7i)22-s + (−3.03e6 − 3.03e6i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.311 − 0.950i)5-s + (−0.796 − 0.796i)7-s + (−0.250 − 0.250i)8-s + (−0.631 − 0.319i)10-s − 1.74i·11-s + (0.605 − 0.605i)13-s − 0.796·14-s − 0.250·16-s + (0.477 − 0.477i)17-s − 0.358i·19-s + (−0.475 + 0.155i)20-s + (−0.872 − 0.872i)22-s + (−0.0984 − 0.0984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.644 - 0.764i$
Analytic conductor: \(69.1508\)
Root analytic conductor: \(8.31570\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :11/2),\ -0.644 - 0.764i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.765427 + 1.64725i\)
\(L(\frac12)\) \(\approx\) \(0.765427 + 1.64725i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-22.6 + 22.6i)T \)
3 \( 1 \)
5 \( 1 + (2.17e3 + 6.63e3i)T \)
good7 \( 1 + (3.54e4 + 3.54e4i)T + 1.97e9iT^{2} \)
11 \( 1 + 9.32e5iT - 2.85e11T^{2} \)
13 \( 1 + (-8.10e5 + 8.10e5i)T - 1.79e12iT^{2} \)
17 \( 1 + (-2.79e6 + 2.79e6i)T - 3.42e13iT^{2} \)
19 \( 1 + 3.87e6iT - 1.16e14T^{2} \)
23 \( 1 + (3.03e6 + 3.03e6i)T + 9.52e14iT^{2} \)
29 \( 1 + 7.24e6T + 1.22e16T^{2} \)
31 \( 1 - 1.58e8T + 2.54e16T^{2} \)
37 \( 1 + (-1.11e8 - 1.11e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 6.52e8iT - 5.50e17T^{2} \)
43 \( 1 + (1.04e9 - 1.04e9i)T - 9.29e17iT^{2} \)
47 \( 1 + (-1.50e9 + 1.50e9i)T - 2.47e18iT^{2} \)
53 \( 1 + (-1.36e9 - 1.36e9i)T + 9.26e18iT^{2} \)
59 \( 1 + 5.53e9T + 3.01e19T^{2} \)
61 \( 1 + 4.03e9T + 4.35e19T^{2} \)
67 \( 1 + (7.47e9 + 7.47e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 2.31e10iT - 2.31e20T^{2} \)
73 \( 1 + (-2.14e10 + 2.14e10i)T - 3.13e20iT^{2} \)
79 \( 1 - 3.06e10iT - 7.47e20T^{2} \)
83 \( 1 + (-4.36e10 - 4.36e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 7.55e10T + 2.77e21T^{2} \)
97 \( 1 + (8.01e10 + 8.01e10i)T + 7.15e21iT^{2} \)
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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

−11.29522841338722468537073137795, −10.29774112114638708461967660684, −9.065376951169695700842762529676, −7.985679534244163306298758224077, −6.35813394506380174470870185038, −5.27580847460010615025394158060, −3.89174067637517358348766496675, −3.06161910828357237195154400016, −1.00149504684796495440423766091, −0.43437864283314760713670858685, 2.03209838456890280029304857042, 3.24854776391366745130856421670, 4.42053261910209359663731733654, 5.99452300645199815513768617692, 6.79599239003134557599458295887, 7.85882462942819679665374461569, 9.325292086562860215168866143682, 10.36523272833496553770443626904, 11.84571582635888907734367767948, 12.48171506550323070743475955569

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