↑

Properties

Label 2-240-15.14-c8-0-40
Degree $2$
Conductor $240$
Sign $0.480 + 0.876i$
Analytic cond. $97.7708$
Root an. cond. $9.88791$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−76.6 − 26.2i)3-s + (−461. − 420. i)5-s − 982. i·7-s + (5.18e3 + 4.02e3i)9-s − 8.65e3i·11-s − 1.81e4i·13-s + (2.43e4 + 4.43e4i)15-s − 4.88e3·17-s + 7.35e4·19-s + (−2.57e4 + 7.52e4i)21-s − 1.16e4·23-s + (3.61e4 + 3.88e5i)25-s + (−2.91e5 − 4.44e5i)27-s + 9.81e5i·29-s + 8.53e5·31-s + ⋯
L(s)  = 1  + (−0.945 − 0.324i)3-s + (−0.739 − 0.673i)5-s − 0.409i·7-s + (0.789 + 0.613i)9-s − 0.591i·11-s − 0.636i·13-s + (0.480 + 0.876i)15-s − 0.0584·17-s + 0.564·19-s + (−0.132 + 0.386i)21-s − 0.0416·23-s + (0.0925 + 0.995i)25-s + (−0.548 − 0.836i)27-s + 1.38i·29-s + 0.923·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.480 + 0.876i$
Analytic conductor: \(97.7708\)
Root analytic conductor: \(9.88791\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :4),\ 0.480 + 0.876i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.152959125\)
\(L(\frac12)\) \(\approx\) \(1.152959125\)
\(L(5)\) 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 \( 1 + (76.6 + 26.2i)T \)
5 \( 1 + (461. + 420. i)T \)
good7 \( 1 + 982. iT - 5.76e6T^{2} \)
11 \( 1 + 8.65e3iT - 2.14e8T^{2} \)
13 \( 1 + 1.81e4iT - 8.15e8T^{2} \)
17 \( 1 + 4.88e3T + 6.97e9T^{2} \)
19 \( 1 - 7.35e4T + 1.69e10T^{2} \)
23 \( 1 + 1.16e4T + 7.83e10T^{2} \)
29 \( 1 - 9.81e5iT - 5.00e11T^{2} \)
31 \( 1 - 8.53e5T + 8.52e11T^{2} \)
37 \( 1 - 7.12e5iT - 3.51e12T^{2} \)
41 \( 1 - 4.65e6iT - 7.98e12T^{2} \)
43 \( 1 - 2.20e6iT - 1.16e13T^{2} \)
47 \( 1 - 3.60e6T + 2.38e13T^{2} \)
53 \( 1 + 1.28e7T + 6.22e13T^{2} \)
59 \( 1 + 1.13e7iT - 1.46e14T^{2} \)
61 \( 1 - 8.48e6T + 1.91e14T^{2} \)
67 \( 1 + 9.68e6iT - 4.06e14T^{2} \)
71 \( 1 + 1.68e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.32e7iT - 8.06e14T^{2} \)
79 \( 1 - 1.15e7T + 1.51e15T^{2} \)
83 \( 1 - 7.18e7T + 2.25e15T^{2} \)
89 \( 1 - 7.55e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.21e8iT - 7.83e15T^{2} \)
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

−10.79872780490505043606034053275, −9.682310476255838830506948886154, −8.343199330427643041468092793428, −7.58666879858329976863285018529, −6.50110763295826170860918834115, −5.35087095398535855678304282393, −4.52096100199784039020203253556, −3.22953065862630126514454424101, −1.29471486899785580973595215852, −0.56054089297980006249734846330, 0.60794194310372017189181635047, 2.25290803949560324745416028536, 3.74627819936492608515724001888, 4.62696927249336461585351735567, 5.85552098723324905491390682979, 6.82498512175263041802111968815, 7.67374164863745777361484048638, 9.086036615243937696934087777128, 10.09150680076373348610584688513, 10.91015054063972388841273589132

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