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Properties

Label 2-90-45.4-c1-0-0
Degree $2$
Conductor $90$
Sign $-0.401 - 0.915i$
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 + (−1.30 + 1.81i)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 + (−2.92 − 2.54i)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 + (−0.584 + 0.811i)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 + (−0.754 − 0.656i)15-s + (−0.125 − 0.216i)16-s − 1.43i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.401 - 0.915i$
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: \((2,\ 90,\ (\ :1/2),\ -0.401 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.359965 + 0.551162i\)
\(L(\frac12)\) \(\approx\) \(0.359965 + 0.551162i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.30 - 1.81i)T \)
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

−14.80252802390489462235161184135, −13.80910749006305444922646164171, −11.73164429480254117775794839483, −11.22838965258597206306847151996, −9.862181415399079699869606501466, −9.236603134252167103375156632271, −7.68032047276718662452747175486, −6.56722115902533300852606747793, −4.91569702126479824903541430576, −3.25308499998496659481242528402, 1.13154269407539701542357772075, 3.49130015533201372340507188095, 5.69972505269856589164084678426, 7.15732078721980594169475192652, 8.339848874134542507746416653377, 8.975974143258665538829426203126, 10.80949910862099754801095379361, 11.65291547592846516934788169979, 12.68020831287898065998586212391, 13.34870993727116602127202696599

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