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Properties

Label 2-90-45.4-c1-0-3
Degree $2$
Conductor $90$
Sign $0.687 + 0.726i$
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 + (0.866 − 1.5i)3-s + (0.499 − 0.866i)4-s + (−1.23 − 1.86i)5-s + 1.73i·6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (−1.5 − 2.59i)9-s + (2 + i)10-s + (1 + 1.73i)11-s + (−0.866 − 1.49i)12-s + (5.19 + 3i)13-s + (−0.499 + 0.866i)14-s + (−3.86 + 0.232i)15-s + (−0.5 − 0.866i)16-s + 2i·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 − 0.866i)3-s + (0.249 − 0.433i)4-s + (−0.550 − 0.834i)5-s + 0.707i·6-s + (0.327 − 0.188i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (0.632 + 0.316i)10-s + (0.301 + 0.522i)11-s + (−0.250 − 0.433i)12-s + (1.44 + 0.832i)13-s + (−0.133 + 0.231i)14-s + (−0.998 + 0.0599i)15-s + (−0.125 − 0.216i)16-s + 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.687 + 0.726i$
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: \((2,\ 90,\ (\ :1/2),\ 0.687 + 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763323 - 0.328462i\)
\(L(\frac12)\) \(\approx\) \(0.763323 - 0.328462i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 + (1.23 + 1.86i)T \)
good7 \( 1 + (-0.866 + 0.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.19 - 3i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + (-5.5 + 9.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.46 - 2i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.06 + 3.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.52 + 5.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.52 - 5.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 + (-6.92 + 4i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.03304208730811811662035362915, −12.90677665677312781423702998977, −11.95936603095552567149508950768, −10.76900444268447423359818747116, −8.940727242928583719450310667647, −8.546790256148184901921862613776, −7.33029502415470965408880901826, −6.17352473400229918546252539018, −4.15060236543285869177240127652, −1.54104431390116796810643042806, 2.87024766510310283335615375952, 4.11216164284346614745955484892, 6.21852382937063131289599362212, 8.008583585967385544526659953745, 8.616745062734813337349643340521, 10.05818310010461342315650110470, 10.91337465933609946043284249238, 11.60145159369364902755904963406, 13.31183538077948618598208038245, 14.43800165552648640151433808510

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