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Properties

Label 2-90-5.3-c10-0-0
Degree $2$
Conductor $90$
Sign $-0.830 + 0.557i$
Analytic cond. $57.1821$
Root an. cond. $7.56188$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−16 + 16i)2-s − 512i·4-s + (−2.92e3 + 1.10e3i)5-s + (6.95e3 − 6.95e3i)7-s + (8.19e3 + 8.19e3i)8-s + (2.92e4 − 6.44e4i)10-s − 7.52e4·11-s + (1.09e5 + 1.09e5i)13-s + 2.22e5i·14-s − 2.62e5·16-s + (1.52e6 − 1.52e6i)17-s + 4.03e6i·19-s + (5.63e5 + 1.49e6i)20-s + (1.20e6 − 1.20e6i)22-s + (7.12e5 + 7.12e5i)23-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 0.5i·4-s + (−0.936 + 0.352i)5-s + (0.413 − 0.413i)7-s + (0.250 + 0.250i)8-s + (0.292 − 0.644i)10-s − 0.467·11-s + (0.295 + 0.295i)13-s + 0.413i·14-s − 0.250·16-s + (1.07 − 1.07i)17-s + 1.63i·19-s + (0.176 + 0.468i)20-s + (0.233 − 0.233i)22-s + (0.110 + 0.110i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.830 + 0.557i$
Analytic conductor: \(57.1821\)
Root analytic conductor: \(7.56188\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :5),\ -0.830 + 0.557i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0133462 - 0.0438010i\)
\(L(\frac12)\) \(\approx\) \(0.0133462 - 0.0438010i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (16 - 16i)T \)
3 \( 1 \)
5 \( 1 + (2.92e3 - 1.10e3i)T \)
good7 \( 1 + (-6.95e3 + 6.95e3i)T - 2.82e8iT^{2} \)
11 \( 1 + 7.52e4T + 2.59e10T^{2} \)
13 \( 1 + (-1.09e5 - 1.09e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (-1.52e6 + 1.52e6i)T - 2.01e12iT^{2} \)
19 \( 1 - 4.03e6iT - 6.13e12T^{2} \)
23 \( 1 + (-7.12e5 - 7.12e5i)T + 4.14e13iT^{2} \)
29 \( 1 - 4.46e5iT - 4.20e14T^{2} \)
31 \( 1 + 2.90e7T + 8.19e14T^{2} \)
37 \( 1 + (9.11e5 - 9.11e5i)T - 4.80e15iT^{2} \)
41 \( 1 - 1.63e8T + 1.34e16T^{2} \)
43 \( 1 + (-1.18e8 - 1.18e8i)T + 2.16e16iT^{2} \)
47 \( 1 + (2.76e8 - 2.76e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (3.08e8 + 3.08e8i)T + 1.74e17iT^{2} \)
59 \( 1 + 9.40e8iT - 5.11e17T^{2} \)
61 \( 1 + 1.35e9T + 7.13e17T^{2} \)
67 \( 1 + (-8.53e8 + 8.53e8i)T - 1.82e18iT^{2} \)
71 \( 1 + 2.82e9T + 3.25e18T^{2} \)
73 \( 1 + (2.75e9 + 2.75e9i)T + 4.29e18iT^{2} \)
79 \( 1 + 3.32e9iT - 9.46e18T^{2} \)
83 \( 1 + (1.34e9 + 1.34e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 2.66e9iT - 3.11e19T^{2} \)
97 \( 1 + (5.26e8 - 5.26e8i)T - 7.37e19iT^{2} \)
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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

−12.59047835038335243053094632440, −11.43905507125841002723722711154, −10.56511900165902474088918688501, −9.376830612286420976622115212732, −7.87030358784001734511219980935, −7.55620598913529365511422264103, −6.03978299208710125706850888295, −4.61738010846223113175792874356, −3.24239914921026275585713059039, −1.34733420686026918864130563558, 0.01585557333013819657332610734, 1.24853826842966440705034207192, 2.81331673473576785832641072777, 4.10277754027071764886927567336, 5.45178147387614648758159748437, 7.29082244531880682872663899066, 8.241046563618072868740749422948, 9.081243177389833155716191283070, 10.54977695029857447415687966038, 11.36282689371420901992131283118

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