↑

Properties

Label 2-240-15.14-c2-0-9
Degree $2$
Conductor $240$
Sign $0.999 + 0.0169i$
Analytic cond. $6.53952$
Root an. cond. $2.55724$
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  + (−2.49 − 1.67i)3-s + (4.19 − 2.71i)5-s + 12.7i·7-s + (3.41 + 8.32i)9-s − 12.6i·11-s + 7.44i·13-s + (−14.9 − 0.254i)15-s + 14.0·17-s + 31.0·19-s + (21.3 − 31.8i)21-s + 7.50·23-s + (10.2 − 22.7i)25-s + (5.40 − 26.4i)27-s + 15.7i·29-s + 20.4·31-s + ⋯
L(s)  = 1  + (−0.830 − 0.557i)3-s + (0.839 − 0.542i)5-s + 1.82i·7-s + (0.379 + 0.925i)9-s − 1.14i·11-s + 0.572i·13-s + (−0.999 − 0.0169i)15-s + 0.826·17-s + 1.63·19-s + (1.01 − 1.51i)21-s + 0.326·23-s + (0.410 − 0.911i)25-s + (0.200 − 0.979i)27-s + 0.542i·29-s + 0.660·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.999 + 0.0169i$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ 0.999 + 0.0169i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.43517 - 0.0121605i\)
\(L(\frac12)\) \(\approx\) \(1.43517 - 0.0121605i\)
\(L(2)\) 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 \( 1 + (2.49 + 1.67i)T \)
5 \( 1 + (-4.19 + 2.71i)T \)
good7 \( 1 - 12.7iT - 49T^{2} \)
11 \( 1 + 12.6iT - 121T^{2} \)
13 \( 1 - 7.44iT - 169T^{2} \)
17 \( 1 - 14.0T + 289T^{2} \)
19 \( 1 - 31.0T + 361T^{2} \)
23 \( 1 - 7.50T + 529T^{2} \)
29 \( 1 - 15.7iT - 841T^{2} \)
31 \( 1 - 20.4T + 961T^{2} \)
37 \( 1 - 12.9iT - 1.36e3T^{2} \)
41 \( 1 + 13.8iT - 1.68e3T^{2} \)
43 \( 1 - 30.0iT - 1.84e3T^{2} \)
47 \( 1 - 20.2T + 2.20e3T^{2} \)
53 \( 1 + 29.1T + 2.80e3T^{2} \)
59 \( 1 + 47.6iT - 3.48e3T^{2} \)
61 \( 1 - 43.0T + 3.72e3T^{2} \)
67 \( 1 - 0.630iT - 4.48e3T^{2} \)
71 \( 1 - 90.4iT - 5.04e3T^{2} \)
73 \( 1 + 46.2iT - 5.32e3T^{2} \)
79 \( 1 + 37.9T + 6.24e3T^{2} \)
83 \( 1 + 80.2T + 6.88e3T^{2} \)
89 \( 1 - 140. iT - 7.92e3T^{2} \)
97 \( 1 - 10.3iT - 9.40e3T^{2} \)
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

−11.94659785282797265874146189442, −11.28132978835279478110444936590, −9.871452038154895205593935779458, −9.003342327124942207383496129140, −8.069389549974261018330817090951, −6.51351799995497270334040120070, −5.62032871157817876247777333641, −5.17960307199832557281817218358, −2.78208952184208912452297363544, −1.30395879449420470822513426213, 1.08080023576183287156607738105, 3.36515458633466123328605587338, 4.60395881599990879520137037377, 5.67768089500894939015353537461, 6.93866023622742634365251807248, 7.52966953730245175161039225327, 9.624218421932112412423148303942, 10.05565041710390194508011769444, 10.68888901732121019610666827033, 11.72940471000698725985837991920

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