↑

Properties

Label 2-2520-840.59-c0-0-6
Degree $2$
Conductor $2520$
Sign $-0.999 - 0.0431i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.707 − 0.707i)7-s + 0.999i·8-s + (0.866 + 0.499i)10-s + (−0.448 − 0.258i)11-s − 1.93i·13-s + (0.965 + 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)19-s − 0.999·20-s + 0.517·22-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.965 + 1.67i)26-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.707 − 0.707i)7-s + 0.999i·8-s + (0.866 + 0.499i)10-s + (−0.448 − 0.258i)11-s − 1.93i·13-s + (0.965 + 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)19-s − 0.999·20-s + 0.517·22-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.965 + 1.67i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.999 - 0.0431i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.999 - 0.0431i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1541896512\)
\(L(\frac12)\) \(\approx\) \(0.1541896512\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.93iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + 0.517T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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

−8.417999007765749515096617406453, −7.983046594767630421045149449992, −7.57362403004777823815086223848, −6.38178118112275484337031252679, −5.78103023624020710263935122403, −4.95690558064563698008449152782, −3.86917804985841868341860292505, −2.85492217810657940321506184939, −1.32282367428908738240996956367, −0.13582808960997706155109265551, 2.18427489676889805561817834551, 2.47196114522602099103383422768, 3.79471641459794778772694857529, 4.31766302391634577695401882519, 5.97137640106841805237740299726, 6.67431649265759575141372654150, 7.15669526572573724566832673945, 8.114120179590897501564808655457, 8.832964883092729035861564188597, 9.447748013620951530024054533180

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