Properties

Label 2-90-5.4-c3-0-2
Degree $2$
Conductor $90$
Sign $0.894 - 0.447i$
Analytic cond. $5.31017$
Root an. cond. $2.30438$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 4·4-s + (5 + 10i)5-s + 26i·7-s + 8i·8-s + (20 − 10i)10-s + 28·11-s − 12i·13-s + 52·14-s + 16·16-s + 64i·17-s + 60·19-s + (−20 − 40i)20-s − 56i·22-s − 58i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.447 + 0.894i)5-s + 1.40i·7-s + 0.353i·8-s + (0.632 − 0.316i)10-s + 0.767·11-s − 0.256i·13-s + 0.992·14-s + 0.250·16-s + 0.913i·17-s + 0.724·19-s + (−0.223 − 0.447i)20-s − 0.542i·22-s − 0.525i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(5.31017\)
Root analytic conductor: \(2.30438\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.47223 + 0.347546i\)
\(L(\frac12)\) \(\approx\) \(1.47223 + 0.347546i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 \)
5 \( 1 + (-5 - 10i)T \)
good7 \( 1 - 26iT - 343T^{2} \)
11 \( 1 - 28T + 1.33e3T^{2} \)
13 \( 1 + 12iT - 2.19e3T^{2} \)
17 \( 1 - 64iT - 4.91e3T^{2} \)
19 \( 1 - 60T + 6.85e3T^{2} \)
23 \( 1 + 58iT - 1.21e4T^{2} \)
29 \( 1 - 90T + 2.43e4T^{2} \)
31 \( 1 + 128T + 2.97e4T^{2} \)
37 \( 1 - 236iT - 5.06e4T^{2} \)
41 \( 1 + 242T + 6.89e4T^{2} \)
43 \( 1 + 362iT - 7.95e4T^{2} \)
47 \( 1 + 226iT - 1.03e5T^{2} \)
53 \( 1 + 108iT - 1.48e5T^{2} \)
59 \( 1 + 20T + 2.05e5T^{2} \)
61 \( 1 - 542T + 2.26e5T^{2} \)
67 \( 1 + 434iT - 3.00e5T^{2} \)
71 \( 1 - 1.12e3T + 3.57e5T^{2} \)
73 \( 1 + 632iT - 3.89e5T^{2} \)
79 \( 1 - 720T + 4.93e5T^{2} \)
83 \( 1 + 478iT - 5.71e5T^{2} \)
89 \( 1 + 490T + 7.04e5T^{2} \)
97 \( 1 - 1.45e3iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.66177341479452577170666142652, −12.40460183519336567068946869050, −11.62579111570910799448357892943, −10.48569249280349020229035668800, −9.453491132063162949116773119033, −8.395198648954966982696387793693, −6.58374486232062297849879486054, −5.40522984126788248683214496483, −3.40197252937799934279077142547, −2.03780995823625177150951216368, 1.00308999499348308821136217698, 3.98229243459530304075045869866, 5.19060409633077467208310361828, 6.69141129736587027760913977770, 7.73717373385852751475773673662, 9.121979813686114628549284665492, 9.931513698976302874941609123232, 11.44230959184334270178854208085, 12.78133766531299122438211706657, 13.79934501469412659492758232011

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