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Properties

Label 2-90-45.34-c1-0-5
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: $\chi_{90} (79, \cdot )$
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

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

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