Properties

Label 2-30e2-4.3-c2-0-55
Degree $2$
Conductor $900$
Sign $0.984 - 0.176i$
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  + (0.177 + 1.99i)2-s + (−3.93 + 0.707i)4-s − 1.19i·7-s + (−2.10 − 7.71i)8-s + 8.22i·11-s + 11.1·13-s + (2.38 − 0.212i)14-s + (14.9 − 5.57i)16-s − 20.9·17-s − 27.9i·19-s + (−16.3 + 1.46i)22-s − 9.48i·23-s + (1.98 + 22.2i)26-s + (0.845 + 4.70i)28-s − 40.4·29-s + ⋯
L(s)  = 1  + (0.0888 + 0.996i)2-s + (−0.984 + 0.176i)4-s − 0.170i·7-s + (−0.263 − 0.964i)8-s + 0.747i·11-s + 0.860·13-s + (0.170 − 0.0151i)14-s + (0.937 − 0.348i)16-s − 1.23·17-s − 1.47i·19-s + (−0.744 + 0.0663i)22-s − 0.412i·23-s + (0.0764 + 0.857i)26-s + (0.0302 + 0.168i)28-s − 1.39·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.984 - 0.176i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.984 - 0.176i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.437826455\)
\(L(\frac12)\) \(\approx\) \(1.437826455\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.177 - 1.99i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 1.19iT - 49T^{2} \)
11 \( 1 - 8.22iT - 121T^{2} \)
13 \( 1 - 11.1T + 169T^{2} \)
17 \( 1 + 20.9T + 289T^{2} \)
19 \( 1 + 27.9iT - 361T^{2} \)
23 \( 1 + 9.48iT - 529T^{2} \)
29 \( 1 + 40.4T + 841T^{2} \)
31 \( 1 + 55.3iT - 961T^{2} \)
37 \( 1 - 50.1T + 1.36e3T^{2} \)
41 \( 1 - 73.6T + 1.68e3T^{2} \)
43 \( 1 - 19.0iT - 1.84e3T^{2} \)
47 \( 1 - 18.0iT - 2.20e3T^{2} \)
53 \( 1 - 57.2T + 2.80e3T^{2} \)
59 \( 1 + 60.6iT - 3.48e3T^{2} \)
61 \( 1 + 21.3T + 3.72e3T^{2} \)
67 \( 1 + 9.68iT - 4.48e3T^{2} \)
71 \( 1 + 68.6iT - 5.04e3T^{2} \)
73 \( 1 - 84.7T + 5.32e3T^{2} \)
79 \( 1 + 23.2iT - 6.24e3T^{2} \)
83 \( 1 - 93.2iT - 6.88e3T^{2} \)
89 \( 1 + 62.9T + 7.92e3T^{2} \)
97 \( 1 - 91.3T + 9.40e3T^{2} \)
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

−9.437286851997241233721155788061, −9.211152400463330250787086762137, −8.085527511824319192461838765264, −7.33620063353616142831066985711, −6.53175583327073256205800677473, −5.75161718043559862796118528629, −4.55976112899849350871636297711, −4.01977439666805730082506154008, −2.42386077253486764996228700649, −0.54971279035517700630954197288, 1.08324463232068237624997266384, 2.28363781346120824269704011913, 3.50087652761764012466058253561, 4.19383896851722096997065685032, 5.50338445820076749481193003712, 6.11940602286046465194786503757, 7.53694209639330906573462255537, 8.604527386755822193967283433315, 9.001329594475034341619017831209, 10.05191530717553129511478170795

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