↑

Properties

Label 2-90-45.4-c1-0-2
Degree $2$
Conductor $90$
Sign $0.993 - 0.114i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.866 + 1.5i)3-s + (0.499 − 0.866i)4-s + (2.23 + 0.133i)5-s + 1.73i·6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (−1.5 − 2.59i)9-s + (1.99 − 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 + (−2.13 + 3.23i)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 + (0.998 + 0.0599i)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 + (−0.550 + 0.834i)15-s + (−0.125 − 0.216i)16-s − 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.993 - 0.114i$
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: \((2,\ 90,\ (\ :1/2),\ 0.993 - 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22505 + 0.0702613i\)
\(L(\frac12)\) \(\approx\) \(1.22505 + 0.0702613i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.133i)T \)
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 \( 1 + (6.92 - 4i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.35544749301079013769529954766, −12.85295718903619488271351804739, −12.19062534326430898163797428721, −10.73168638131145457989544693944, −10.02758857383745713976956920938, −9.107794971338959898382363204612, −6.82043002097545461274690869636, −5.60810404130365919907561901899, −4.60762477961140238604786618984, −2.73906414150220695599472359919, 2.31534766162471871481446780922, 4.72530765235595003728604336779, 6.14906564400756608464136136607, 6.75993903544224902772219265772, 8.254375647433727327560887201038, 9.779747887727566840538168300942, 11.13066402154609963959697028987, 12.32381791215817728156578994821, 13.06284365441183633661331090682, 13.98216398320997143553975093400

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