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Properties

Label 2-90-9.4-c1-0-1
Degree $2$
Conductor $90$
Sign $0.283 - 0.959i$
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 + (0.5 + 1.65i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−1.18 + 1.26i)6-s + (−1.68 − 2.92i)7-s − 0.999·8-s + (−2.5 + 1.65i)9-s + 0.999·10-s + (2.18 + 3.78i)11-s + (−1.68 − 0.396i)12-s + (3.37 − 5.84i)13-s + (1.68 − 2.92i)14-s + (1.68 + 0.396i)15-s + (−0.5 − 0.866i)16-s − 1.62·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.957i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.484 + 0.515i)6-s + (−0.637 − 1.10i)7-s − 0.353·8-s + (−0.833 + 0.552i)9-s + 0.316·10-s + (0.659 + 1.14i)11-s + (−0.486 − 0.114i)12-s + (0.935 − 1.61i)13-s + (0.450 − 0.780i)14-s + (0.435 + 0.102i)15-s + (−0.125 − 0.216i)16-s − 0.394·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.283 - 0.959i$
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: \((2,\ 90,\ (\ :1/2),\ 0.283 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.952687 + 0.712147i\)
\(L(\frac12)\) \(\approx\) \(0.952687 + 0.712147i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 1.65i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (1.68 + 2.92i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.37 + 5.84i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.62T + 17T^{2} \)
19 \( 1 + 2.37T + 19T^{2} \)
23 \( 1 + (0.686 - 1.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.686 + 1.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.37 - 4.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.81 - 4.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.68 - 6.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + (2.18 - 3.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.05 - 7.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 3.11T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.68 + 6.38i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + (-4.18 - 7.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.43490771502679653050059736033, −13.42957053926697582185936711868, −12.59639790250329447897291874680, −10.82610090419855109496468636602, −9.929401347418033588231230650898, −8.827491123575263196328435927919, −7.53902124012673686328720102525, −6.07350551737081185459904622563, −4.61754204441517923884302498967, −3.52691557687210991403877811843, 2.09395855941093279346428341933, 3.58020325690293215600869944148, 5.97758986494344774922040568308, 6.59889260971893387586052129786, 8.666885025426302522464385467317, 9.240700962235215870810921345721, 11.11391499915297159034028155745, 11.80750156771041678840283505239, 12.86852272375675210236324786536, 13.80499442259820274735405722680

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