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Properties

Label 2-90-5.4-c11-0-10
Degree $2$
Conductor $90$
Sign $0.914 + 0.404i$
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 + (−2.82e3 + 6.39e3i)5-s − 1.59e4i·7-s + 3.27e4i·8-s + (2.04e5 + 9.04e4i)10-s − 6.23e5·11-s − 1.54e6i·13-s − 5.09e5·14-s + 1.04e6·16-s + 1.11e6i·17-s − 1.45e7·19-s + (2.89e6 − 6.54e6i)20-s + 1.99e7i·22-s + 6.03e6i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.404 + 0.914i)5-s − 0.357i·7-s + 0.353i·8-s + (0.646 + 0.286i)10-s − 1.16·11-s − 1.15i·13-s − 0.252·14-s + 0.250·16-s + 0.191i·17-s − 1.35·19-s + (0.202 − 0.457i)20-s + 0.825i·22-s + 0.195i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.914 + 0.404i$
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: \((2,\ 90,\ (\ :11/2),\ 0.914 + 0.404i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.16842 - 0.246908i\)
\(L(\frac12)\) \(\approx\) \(1.16842 - 0.246908i\)
\(L(\frac{13}{2})\) 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 \( 1 + (2.82e3 - 6.39e3i)T \)
good7 \( 1 + 1.59e4iT - 1.97e9T^{2} \)
11 \( 1 + 6.23e5T + 2.85e11T^{2} \)
13 \( 1 + 1.54e6iT - 1.79e12T^{2} \)
17 \( 1 - 1.11e6iT - 3.42e13T^{2} \)
19 \( 1 + 1.45e7T + 1.16e14T^{2} \)
23 \( 1 - 6.03e6iT - 9.52e14T^{2} \)
29 \( 1 + 2.91e7T + 1.22e16T^{2} \)
31 \( 1 - 2.33e8T + 2.54e16T^{2} \)
37 \( 1 - 6.65e8iT - 1.77e17T^{2} \)
41 \( 1 - 6.63e8T + 5.50e17T^{2} \)
43 \( 1 - 4.11e8iT - 9.29e17T^{2} \)
47 \( 1 + 2.47e9iT - 2.47e18T^{2} \)
53 \( 1 - 3.69e9iT - 9.26e18T^{2} \)
59 \( 1 - 1.25e9T + 3.01e19T^{2} \)
61 \( 1 + 4.05e9T + 4.35e19T^{2} \)
67 \( 1 - 1.84e10iT - 1.22e20T^{2} \)
71 \( 1 + 3.19e9T + 2.31e20T^{2} \)
73 \( 1 + 1.51e10iT - 3.13e20T^{2} \)
79 \( 1 - 4.26e10T + 7.47e20T^{2} \)
83 \( 1 + 5.86e10iT - 1.28e21T^{2} \)
89 \( 1 - 3.40e10T + 2.77e21T^{2} \)
97 \( 1 + 1.37e11iT - 7.15e21T^{2} \)
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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

−11.68022891922439786770576595133, −10.48171354726038007470424017628, −10.28526034127107706605763147398, −8.422914081393796723201824029087, −7.53853378357779742605972480733, −6.06955932222089353747226983700, −4.57740009242435411928897645595, −3.26900967114692319043395509563, −2.36514707007973775794879101515, −0.59582743890034101072970743803, 0.51405972189965656966632650225, 2.22012095383061859640036446526, 4.12004518561818957646215864295, 5.03122504460321889433494359031, 6.24598714782911183827716429109, 7.62008495457916843132862946357, 8.546536703180808348690175818617, 9.424131473332028032107354329543, 10.88345114458000165656779358587, 12.21741708182473592784134669837

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