↑

Properties

Label 2-240-80.69-c1-0-0
Degree $2$
Conductor $240$
Sign $-0.880 + 0.473i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
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  + (0.550 + 1.30i)2-s + (−0.707 − 0.707i)3-s + (−1.39 + 1.43i)4-s + (−2.23 + 0.162i)5-s + (0.531 − 1.31i)6-s − 2.93·7-s + (−2.63 − 1.02i)8-s + 1.00i·9-s + (−1.43 − 2.81i)10-s + (−0.663 − 0.663i)11-s + (1.99 − 0.0281i)12-s + (−1.12 − 1.12i)13-s + (−1.61 − 3.82i)14-s + (1.69 + 1.46i)15-s + (−0.112 − 3.99i)16-s + 7.47i·17-s + ⋯
L(s)  = 1  + (0.389 + 0.921i)2-s + (−0.408 − 0.408i)3-s + (−0.697 + 0.716i)4-s + (−0.997 + 0.0724i)5-s + (0.217 − 0.534i)6-s − 1.10·7-s + (−0.931 − 0.363i)8-s + 0.333i·9-s + (−0.454 − 0.890i)10-s + (−0.200 − 0.200i)11-s + (0.577 − 0.00813i)12-s + (−0.312 − 0.312i)13-s + (−0.431 − 1.02i)14-s + (0.436 + 0.377i)15-s + (−0.0281 − 0.999i)16-s + 1.81i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.880 + 0.473i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.880 + 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0595753 - 0.236579i\)
\(L(\frac12)\) \(\approx\) \(0.0595753 - 0.236579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.550 - 1.30i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (2.23 - 0.162i)T \)
good7 \( 1 + 2.93T + 7T^{2} \)
11 \( 1 + (0.663 + 0.663i)T + 11iT^{2} \)
13 \( 1 + (1.12 + 1.12i)T + 13iT^{2} \)
17 \( 1 - 7.47iT - 17T^{2} \)
19 \( 1 + (-0.423 + 0.423i)T - 19iT^{2} \)
23 \( 1 + 6.17T + 23T^{2} \)
29 \( 1 + (2.95 - 2.95i)T - 29iT^{2} \)
31 \( 1 - 1.82T + 31T^{2} \)
37 \( 1 + (-5.53 + 5.53i)T - 37iT^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
43 \( 1 + (0.897 - 0.897i)T - 43iT^{2} \)
47 \( 1 + 4.12iT - 47T^{2} \)
53 \( 1 + (0.146 - 0.146i)T - 53iT^{2} \)
59 \( 1 + (7.72 + 7.72i)T + 59iT^{2} \)
61 \( 1 + (7.37 - 7.37i)T - 61iT^{2} \)
67 \( 1 + (8.68 + 8.68i)T + 67iT^{2} \)
71 \( 1 - 8.95iT - 71T^{2} \)
73 \( 1 + 0.174T + 73T^{2} \)
79 \( 1 - 3.06T + 79T^{2} \)
83 \( 1 + (-9.18 - 9.18i)T + 83iT^{2} \)
89 \( 1 + 8.71iT - 89T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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

−12.68387477703780253756272538553, −12.15638407803474793591257114024, −10.89256454934910389014014806338, −9.685410660568204565998658924269, −8.333900138997615297472624555609, −7.67542439317920147730261994098, −6.55385155306543043992890304425, −5.84052868240819814641312971617, −4.33423443694604933958647262991, −3.25746160171060681141419038299, 0.17598363074291378742856037647, 2.84229838731846964480311450585, 3.97031117218741798811293508369, 4.94673014370976247602460892062, 6.25750144158304550441092791906, 7.56680092283685938442323330711, 9.122194921828709242482965152541, 9.781681264532001542739053034193, 10.73470979409484621785326784235, 11.88063009876056206768523487816

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