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Properties

Label 2-90-9.4-c1-0-3
Degree $2$
Conductor $90$
Sign $0.993 - 0.112i$
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 + (1.68 − 0.396i)6-s + (1.18 + 2.05i)7-s − 0.999·8-s + (−2.5 − 1.65i)9-s + 0.999·10-s + (−0.686 − 1.18i)11-s + (1.18 + 1.26i)12-s + (−2.37 + 4.10i)13-s + (−1.18 + 2.05i)14-s + (−1.18 − 1.26i)15-s + (−0.5 − 0.866i)16-s − 7.37·17-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 + (0.688 − 0.161i)6-s + (0.448 + 0.776i)7-s − 0.353·8-s + (−0.833 − 0.552i)9-s + 0.316·10-s + (−0.206 − 0.358i)11-s + (0.342 + 0.364i)12-s + (−0.657 + 1.13i)13-s + (−0.317 + 0.549i)14-s + (−0.306 − 0.325i)15-s + (−0.125 − 0.216i)16-s − 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.993 - 0.112i$
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: \((2,\ 90,\ (\ :1/2),\ 0.993 - 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21264 + 0.0682506i\)
\(L(\frac12)\) \(\approx\) \(1.21264 + 0.0682506i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 - 2.05i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.686 + 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.37 - 4.10i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.37T + 17T^{2} \)
19 \( 1 - 3.37T + 19T^{2} \)
23 \( 1 + (-2.18 + 3.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.18 - 3.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.37 + 5.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.68 - 9.84i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.813 - 1.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + (-0.686 + 1.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.55 + 7.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.813 + 1.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.11T + 89T^{2} \)
97 \( 1 + (-1.31 - 2.27i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.06624198202556433000129113710, −13.24267135569410288986800778080, −12.23953525672563598223393157501, −11.35904924905659229441383241833, −9.174893344701259403447161937674, −8.519449275222856879777627977821, −7.18561928284831475828817367708, −6.11330580916762578181092999599, −4.72182332536051184038693897135, −2.39556048339853099161668891949, 2.73333088592281439044043826735, 4.25135754063386504291921233653, 5.37707615780800789661141307922, 7.32427582246690967398255724393, 8.845141070370733696941745656239, 10.13799055989745410394471373098, 10.64806944007082287113704939839, 11.74371907800682810060734374785, 13.30040096781920011926219165864, 14.00064214225990792615221542644

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