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Properties

Label 2-90-9.4-c1-0-0
Degree $2$
Conductor $90$
Sign $-0.173 - 0.984i$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−1.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−1.5 − 0.866i)6-s + (2 + 3.46i)7-s − 0.999·8-s + (1.5 − 2.59i)9-s − 0.999·10-s + (−1.5 − 2.59i)11-s − 1.73i·12-s + (2 − 3.46i)13-s + (−1.99 + 3.46i)14-s − 1.73i·15-s + (−0.5 − 0.866i)16-s + 3·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.866 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.612 − 0.353i)6-s + (0.755 + 1.30i)7-s − 0.353·8-s + (0.5 − 0.866i)9-s − 0.316·10-s + (−0.452 − 0.783i)11-s − 0.499i·12-s + (0.554 − 0.960i)13-s + (−0.534 + 0.925i)14-s − 0.447i·15-s + (−0.125 − 0.216i)16-s + 0.727·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.586896 + 0.699436i\)
\(L(\frac12)\) \(\approx\) \(0.586896 + 0.699436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (5.5 + 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.72207414904648861181222780286, −13.44387070404141502838981215169, −12.03512188012439476819004166954, −11.48312375693963712779572268078, −10.18468736520390327685114226225, −8.720447699403903692913811793550, −7.56100226290993375336673028027, −5.72774957355078234654894370652, −5.43647428024433711196140477545, −3.46176549768886153776646124433, 1.41437517622745033055006512862, 4.18439722996792072623628775866, 5.20126683946082683463131031453, 6.90402087196382098049944472385, 7.976515536428201680759512564997, 9.863092136411138999146787703555, 10.86829575799433565226397703689, 11.68888483653068727306850491361, 12.64443226667816677943178652930, 13.63428711034586025248457800127

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