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Properties

Label 2-30e2-20.19-c2-0-57
Degree $2$
Conductor $900$
Sign $-0.887 + 0.460i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.90 − 0.618i)2-s + (3.23 + 2.35i)4-s − 5.25·7-s + (−4.70 − 6.47i)8-s + 19.9i·11-s + 8.47i·13-s + (9.99 + 3.24i)14-s + (4.94 + 15.2i)16-s − 11.8i·17-s − 15.2i·19-s + (12.3 − 37.8i)22-s − 0.555·23-s + (5.23 − 16.1i)26-s + (−17.0 − 12.3i)28-s − 10.9·29-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s − 0.751·7-s + (−0.587 − 0.809i)8-s + 1.81i·11-s + 0.651i·13-s + (0.714 + 0.232i)14-s + (0.309 + 0.951i)16-s − 0.699i·17-s − 0.800i·19-s + (0.559 − 1.72i)22-s − 0.0241·23-s + (0.201 − 0.619i)26-s + (−0.607 − 0.441i)28-s − 0.377·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.887 + 0.460i$
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: \((2,\ 900,\ (\ :1),\ -0.887 + 0.460i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1506933712\)
\(L(\frac12)\) \(\approx\) \(0.1506933712\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.90 + 0.618i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 5.25T + 49T^{2} \)
11 \( 1 - 19.9iT - 121T^{2} \)
13 \( 1 - 8.47iT - 169T^{2} \)
17 \( 1 + 11.8iT - 289T^{2} \)
19 \( 1 + 15.2iT - 361T^{2} \)
23 \( 1 + 0.555T + 529T^{2} \)
29 \( 1 + 10.9T + 841T^{2} \)
31 \( 1 - 8.29iT - 961T^{2} \)
37 \( 1 + 18.3iT - 1.36e3T^{2} \)
41 \( 1 - 14.5T + 1.68e3T^{2} \)
43 \( 1 + 22.2T + 1.84e3T^{2} \)
47 \( 1 + 53.3T + 2.20e3T^{2} \)
53 \( 1 + 66.3iT - 2.80e3T^{2} \)
59 \( 1 + 17.4iT - 3.48e3T^{2} \)
61 \( 1 - 90.1T + 3.72e3T^{2} \)
67 \( 1 - 50.2T + 4.48e3T^{2} \)
71 \( 1 + 80.7iT - 5.04e3T^{2} \)
73 \( 1 - 5.55iT - 5.32e3T^{2} \)
79 \( 1 + 13.8iT - 6.24e3T^{2} \)
83 \( 1 + 76.2T + 6.88e3T^{2} \)
89 \( 1 + 111.T + 7.92e3T^{2} \)
97 \( 1 + 92.8iT - 9.40e3T^{2} \)
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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

−9.646877890498759332984529609135, −9.000506643917526391905884913193, −7.88391396907213482778967845605, −6.94204613470987794761238512189, −6.66067734021638596318433062305, −5.07714398849144474483821175083, −3.94533233315345884430363816723, −2.72029458406553535689003061424, −1.74671266021176587676445761238, −0.07171840508930801372351664917, 1.18390166086736544379095978964, 2.78683226327264713746826759168, 3.67798752104931594096297049205, 5.50785471890458004652452212216, 6.03623197813838201950003303455, 6.85061374333425360088372297084, 8.104292522220335881232683660531, 8.382648298580394236018355370466, 9.439685393121704290947992931410, 10.15002509276289149534336219894

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