↑

Properties

Label 2-450-9.7-c1-0-15
Degree $2$
Conductor $450$
Sign $-0.350 + 0.936i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.724 − 1.57i)3-s + (−0.499 − 0.866i)4-s + (1.72 + 0.158i)6-s + (2.22 − 3.85i)7-s + 0.999·8-s + (−1.94 + 2.28i)9-s + (−0.724 + 1.25i)11-s + (−1 + 1.41i)12-s + (−1.22 − 2.12i)13-s + (2.22 + 3.85i)14-s + (−0.5 + 0.866i)16-s − 3.89·17-s + (−0.999 − 2.82i)18-s − 0.550·19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.418 − 0.908i)3-s + (−0.249 − 0.433i)4-s + (0.704 + 0.0648i)6-s + (0.840 − 1.45i)7-s + 0.353·8-s + (−0.649 + 0.760i)9-s + (−0.218 + 0.378i)11-s + (−0.288 + 0.408i)12-s + (−0.339 − 0.588i)13-s + (0.594 + 1.02i)14-s + (−0.125 + 0.216i)16-s − 0.945·17-s + (−0.235 − 0.666i)18-s − 0.126·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.350 + 0.936i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.350 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.430688 - 0.621167i\)
\(L(\frac12)\) \(\approx\) \(0.430688 - 0.621167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.724 + 1.57i)T \)
5 \( 1 \)
good7 \( 1 + (-2.22 + 3.85i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.724 - 1.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.22 + 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.89T + 17T^{2} \)
19 \( 1 + 0.550T + 19T^{2} \)
23 \( 1 + (1.44 + 2.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.22 + 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.72 + 6.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.224 + 0.389i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.44T + 53T^{2} \)
59 \( 1 + (-5.62 - 9.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.224 + 0.389i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.72 - 8.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.44T + 71T^{2} \)
73 \( 1 - 4.79T + 73T^{2} \)
79 \( 1 + (-3.67 + 6.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2 - 3.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.73611639447033645585036152253, −10.12152217135710720532738925906, −8.653982855262777870888260946812, −7.78068296773172807877278015548, −7.27316521808653283361045097175, −6.38479558876015069227735080342, −5.19104675153303681867775531253, −4.23789075598187419709776373215, −2.07861640367847672782153745443, −0.54929245140165924090500427610, 2.01416681062050999285421245410, 3.30358056307890579615636616315, 4.70215578047065565234953575363, 5.35826271759240857930203965379, 6.59240516263672329952364722125, 8.184640561266854668093273308701, 8.917554317454108490810140853661, 9.468156957518337615908384671058, 10.69690428230069662010218437141, 11.21897584514948404582397844297

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