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Properties

Label 2-90-45.4-c1-0-5
Degree $2$
Conductor $90$
Sign $0.576 + 0.816i$
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  + (0.866 − 0.5i)2-s + (0.158 − 1.72i)3-s + (0.499 − 0.866i)4-s + (−0.917 + 2.03i)5-s + (−0.724 − 1.57i)6-s + (0.389 − 0.224i)7-s − 0.999i·8-s + (−2.94 − 0.548i)9-s + (0.224 + 2.22i)10-s + (1.72 + 2.98i)11-s + (−1.41 − 0.999i)12-s + (−2.12 − 1.22i)13-s + (0.224 − 0.389i)14-s + (3.37 + 1.90i)15-s + (−0.5 − 0.866i)16-s + 5.89i·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.0917 − 0.995i)3-s + (0.249 − 0.433i)4-s + (−0.410 + 0.911i)5-s + (−0.295 − 0.642i)6-s + (0.147 − 0.0849i)7-s − 0.353i·8-s + (−0.983 − 0.182i)9-s + (0.0710 + 0.703i)10-s + (0.520 + 0.900i)11-s + (−0.408 − 0.288i)12-s + (−0.588 − 0.339i)13-s + (0.0600 − 0.104i)14-s + (0.870 + 0.492i)15-s + (−0.125 − 0.216i)16-s + 1.43i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.576 + 0.816i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :1/2),\ 0.576 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12926 - 0.584859i\)
\(L(\frac12)\) \(\approx\) \(1.12926 - 0.584859i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.158 + 1.72i)T \)
5 \( 1 + (0.917 - 2.03i)T \)
good7 \( 1 + (-0.389 + 0.224i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.72 - 2.98i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.12 + 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.89iT - 17T^{2} \)
19 \( 1 - 5.44T + 19T^{2} \)
23 \( 1 + (5.97 + 3.44i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.775 - 1.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.20 + 1.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.85 + 2.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.55iT - 53T^{2} \)
59 \( 1 + (-6.62 + 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.22 + 3.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.94 - 2.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 + (-3.67 - 6.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.46 - 2i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.10T + 89T^{2} \)
97 \( 1 + (11.2 - 6.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.10785871550535578383461112336, −12.70388573077722447510903210273, −12.06436702077582165151651079696, −11.03608042251989246659065417683, −9.823977150577299661080253324953, −7.964055672745042024316698617247, −7.03359102898768566334571652894, −5.86777420491565887386810951752, −3.87973663604563963223802631918, −2.22143501198063388053769767323, 3.40143802318796413810831655341, 4.71292075716634527105090508002, 5.64602718897363475251973089842, 7.55237684885516684005224754985, 8.804359792607036955070896285410, 9.703768129417084126986628733584, 11.50656761677525870193740618190, 11.88379976874163163997560281822, 13.55270254406507274149979798149, 14.25386858388551989938739815372

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