↑

Properties

Label 2-30e2-20.19-c2-0-31
Degree $2$
Conductor $900$
Sign $0.289 + 0.957i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.169 − 1.99i)2-s + (−3.94 − 0.675i)4-s − 12.3·7-s + (−2.01 + 7.74i)8-s + 11.0i·11-s + 2.82i·13-s + (−2.10 + 24.7i)14-s + (15.0 + 5.32i)16-s + 6.52i·17-s − 27.9i·19-s + (22.0 + 1.87i)22-s + 7.90·23-s + (5.61 + 0.477i)26-s + (48.8 + 8.37i)28-s + 50.7·29-s + ⋯
L(s)  = 1  + (0.0847 − 0.996i)2-s + (−0.985 − 0.168i)4-s − 1.77·7-s + (−0.251 + 0.967i)8-s + 1.00i·11-s + 0.216i·13-s + (−0.150 + 1.76i)14-s + (0.942 + 0.332i)16-s + 0.383i·17-s − 1.47i·19-s + (1.00 + 0.0850i)22-s + 0.343·23-s + (0.216 + 0.0183i)26-s + (1.74 + 0.298i)28-s + 1.74·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.289 + 0.957i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.289 + 0.957i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.124742121\)
\(L(\frac12)\) \(\approx\) \(1.124742121\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.169 + 1.99i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 12.3T + 49T^{2} \)
11 \( 1 - 11.0iT - 121T^{2} \)
13 \( 1 - 2.82iT - 169T^{2} \)
17 \( 1 - 6.52iT - 289T^{2} \)
19 \( 1 + 27.9iT - 361T^{2} \)
23 \( 1 - 7.90T + 529T^{2} \)
29 \( 1 - 50.7T + 841T^{2} \)
31 \( 1 + 36.3iT - 961T^{2} \)
37 \( 1 - 18.9iT - 1.36e3T^{2} \)
41 \( 1 + 5.30T + 1.68e3T^{2} \)
43 \( 1 + 45.5T + 1.84e3T^{2} \)
47 \( 1 - 11.7T + 2.20e3T^{2} \)
53 \( 1 + 41.1iT - 2.80e3T^{2} \)
59 \( 1 + 10.7iT - 3.48e3T^{2} \)
61 \( 1 - 56.1T + 3.72e3T^{2} \)
67 \( 1 - 16.1T + 4.48e3T^{2} \)
71 \( 1 - 66.1iT - 5.04e3T^{2} \)
73 \( 1 - 15.6iT - 5.32e3T^{2} \)
79 \( 1 - 123. iT - 6.24e3T^{2} \)
83 \( 1 - 99.6T + 6.88e3T^{2} \)
89 \( 1 - 101.T + 7.92e3T^{2} \)
97 \( 1 + 127. iT - 9.40e3T^{2} \)
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

−9.838452860117441500781599470629, −9.263662419882984477408439205685, −8.371489823764005145712228615391, −7.00294701131256708000659207522, −6.35570771761182967887500143725, −5.08235524291216001478763377226, −4.16063051935998925588861654137, −3.13443331177402742597648616148, −2.30688149751199091522452175065, −0.61894371333456714200648948844, 0.68351523495761217914757990975, 3.08908079751933368230873299404, 3.65074001744020940761406000386, 5.01603983321779490047798003161, 6.06265117053408285140330242429, 6.45193930608680793921079546347, 7.41111530260082699828294440954, 8.428374103841026362396021434288, 9.065135896927713364458555825588, 9.991992771640798728914608419538

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