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Properties

Label 2-90-9.7-c1-0-0
Degree $2$
Conductor $90$
Sign $0.939 - 0.342i$
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 + 1.73i·6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (1.5 − 2.59i)9-s − 0.999·10-s + (−3 + 5.19i)11-s + (−1.49 − 0.866i)12-s + (−1 − 1.73i)13-s + (0.499 + 0.866i)14-s + (1.5 + 0.866i)15-s + (−0.5 + 0.866i)16-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 + 0.707i·6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s − 0.316·10-s + (−0.904 + 1.56i)11-s + (−0.433 − 0.250i)12-s + (−0.277 − 0.480i)13-s + (0.133 + 0.231i)14-s + (0.387 + 0.223i)15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02091 + 0.180015i\)
\(L(\frac12)\) \(\approx\) \(1.02091 + 0.180015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.56079602071985426039705926516, −13.20930217081189068297682058650, −12.45137356857302849413930973901, −10.50162404784282696573836333329, −9.747830227416623325614311522859, −8.321306931986745331309819886925, −7.48422726612557706065549337647, −6.45954662637191414006009843123, −4.55038068692036260792144174367, −2.33919010819884478167126981392, 2.37537030969457984752294204023, 3.93544610495337274078412802121, 5.57075319935115414868322336336, 7.83575954475383675984806026531, 8.641736633502856368430856265107, 9.612859932227793954121340908248, 10.68583135433577144212687711208, 11.76290228817964208844447116448, 13.27077342351155635351936184337, 13.74390613949249611289264562549

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