↑

Properties

Label 2-450-9.2-c2-0-6
Degree $2$
Conductor $450$
Sign $-0.635 - 0.771i$
Analytic cond. $12.2616$
Root an. cond. $3.50165$
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  + (−1.22 − 0.707i)2-s + (2.44 + 1.73i)3-s + (0.999 + 1.73i)4-s + (−1.77 − 3.85i)6-s + (−4.17 + 7.22i)7-s − 2.82i·8-s + (2.99 + 8.48i)9-s + (0.825 + 0.476i)11-s + (−0.550 + 5.97i)12-s + (−4.84 − 8.39i)13-s + (10.2 − 5.90i)14-s + (−2.00 + 3.46i)16-s + 18.8i·17-s + (2.32 − 12.5i)18-s − 24.6·19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.816 + 0.577i)3-s + (0.249 + 0.433i)4-s + (−0.295 − 0.642i)6-s + (−0.596 + 1.03i)7-s − 0.353i·8-s + (0.333 + 0.942i)9-s + (0.0750 + 0.0433i)11-s + (−0.0458 + 0.497i)12-s + (−0.372 − 0.645i)13-s + (0.730 − 0.421i)14-s + (−0.125 + 0.216i)16-s + 1.11i·17-s + (0.129 − 0.695i)18-s − 1.29·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.635 - 0.771i$
Analytic conductor: \(12.2616\)
Root analytic conductor: \(3.50165\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1),\ -0.635 - 0.771i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.019591276\)
\(L(\frac12)\) \(\approx\) \(1.019591276\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-2.44 - 1.73i)T \)
5 \( 1 \)
good7 \( 1 + (4.17 - 7.22i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-0.825 - 0.476i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (4.84 + 8.39i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 18.8iT - 289T^{2} \)
19 \( 1 + 24.6T + 361T^{2} \)
23 \( 1 + (0.825 - 0.476i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-11.8 - 6.84i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (1.52 + 2.63i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 46.6T + 1.36e3T^{2} \)
41 \( 1 + (9.45 - 5.45i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-22.5 + 39.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (39.2 + 22.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 94.3iT - 2.80e3T^{2} \)
59 \( 1 + (16.2 - 9.39i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (6.54 - 11.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-37.5 - 64.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 18.0iT - 5.04e3T^{2} \)
73 \( 1 - 7.90T + 5.32e3T^{2} \)
79 \( 1 + (-21.8 + 37.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-112. - 65.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 145. iT - 7.92e3T^{2} \)
97 \( 1 + (54.9 - 95.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−10.76775431614507072677383982875, −10.29000384524385931098003378312, −9.307571386695376410607482919238, −8.658564236222432991937827099606, −7.987057128850260275612740075976, −6.68757104293944951677715986114, −5.48618002617282895751075337751, −4.07611028091006669236595430919, −2.98139094443237090647131798319, −2.01579463260259056527562069947, 0.44135118300576617820422473280, 1.99555410392298299120217418502, 3.37471989492607727201229277456, 4.63695789984887851177944707808, 6.42354850096005898140973212555, 6.92381023418840756528500395391, 7.76483623586102061337917499231, 8.715098063416958303161421109425, 9.542707708062155076669674212002, 10.23906100912592042568482607236

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