↑

Properties

Label 2-30e2-45.14-c2-0-28
Degree $2$
Conductor $900$
Sign $-0.886 + 0.462i$
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  + (−2.70 + 1.29i)3-s + (3.00 − 1.73i)7-s + (5.66 − 6.99i)9-s + (−12.3 + 7.14i)11-s + (5.28 + 3.04i)13-s + 11.7·17-s − 23.4·19-s + (−5.88 + 8.56i)21-s + (14.3 − 24.8i)23-s + (−6.30 + 26.2i)27-s + (13.3 − 7.68i)29-s + (−18.4 + 32.0i)31-s + (24.2 − 35.3i)33-s + 46.7i·37-s + (−18.2 − 1.43i)39-s + ⋯
L(s)  = 1  + (−0.902 + 0.430i)3-s + (0.428 − 0.247i)7-s + (0.629 − 0.777i)9-s + (−1.12 + 0.649i)11-s + (0.406 + 0.234i)13-s + 0.690·17-s − 1.23·19-s + (−0.280 + 0.407i)21-s + (0.624 − 1.08i)23-s + (−0.233 + 0.972i)27-s + (0.459 − 0.265i)29-s + (−0.596 + 1.03i)31-s + (0.735 − 1.07i)33-s + 1.26i·37-s + (−0.467 − 0.0368i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 + 0.462i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.886 + 0.462i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.886 + 0.462i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.06212632306\)
\(L(\frac12)\) \(\approx\) \(0.06212632306\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.70 - 1.29i)T \)
5 \( 1 \)
good7 \( 1 + (-3.00 + 1.73i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (12.3 - 7.14i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.28 - 3.04i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 11.7T + 289T^{2} \)
19 \( 1 + 23.4T + 361T^{2} \)
23 \( 1 + (-14.3 + 24.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-13.3 + 7.68i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (18.4 - 32.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 46.7iT - 1.36e3T^{2} \)
41 \( 1 + (17.1 + 9.89i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-4.76 + 2.75i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (5.15 + 8.93i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 41.5T + 2.80e3T^{2} \)
59 \( 1 + (58.5 + 33.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (26.5 + 45.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (27.5 + 15.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 2.28iT - 5.04e3T^{2} \)
73 \( 1 + 86.2iT - 5.32e3T^{2} \)
79 \( 1 + (58.4 + 101. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (65.2 + 113. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 75.3iT - 7.92e3T^{2} \)
97 \( 1 + (24.4 - 14.1i)T + (4.70e3 - 8.14e3i)T^{2} \)
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

−9.804313781449420379252783165632, −8.737825865292629145160710587896, −7.84399972347389397926986082461, −6.84714132995175304299414709747, −6.07227671825097244888764165491, −4.91713795465222872552070286138, −4.52601298336312058393779222072, −3.14167946919504898108818051786, −1.58745936371335995694326049805, −0.02355000349607336906736418209, 1.39414003306919119263806665824, 2.70285882465587404437886948186, 4.12545881870476976063989828832, 5.37325032777366394455106666154, 5.69308009757379849037550142555, 6.81016714048247117604432508929, 7.78402899607287660569318401529, 8.334132462350051823254857057823, 9.523232387650463058620457431682, 10.62410927766507597305791758111

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