Properties

Label 2-1520-1520.189-c0-0-1
Degree $2$
Conductor $1520$
Sign $0.923 + 0.382i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
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  + (−0.195 + 0.980i)2-s + (−1.38 − 1.38i)3-s + (−0.923 − 0.382i)4-s + (0.707 − 0.707i)5-s + (1.63 − 1.08i)6-s + (0.555 − 0.831i)8-s + 2.84i·9-s + (0.555 + 0.831i)10-s + (0.541 − 0.541i)11-s + (0.750 + 1.81i)12-s + (1.17 + 1.17i)13-s − 1.96·15-s + (0.707 + 0.707i)16-s + (−2.79 − 0.555i)18-s + (0.707 + 0.707i)19-s + (−0.923 + 0.382i)20-s + ⋯
L(s)  = 1  + (−0.195 + 0.980i)2-s + (−1.38 − 1.38i)3-s + (−0.923 − 0.382i)4-s + (0.707 − 0.707i)5-s + (1.63 − 1.08i)6-s + (0.555 − 0.831i)8-s + 2.84i·9-s + (0.555 + 0.831i)10-s + (0.541 − 0.541i)11-s + (0.750 + 1.81i)12-s + (1.17 + 1.17i)13-s − 1.96·15-s + (0.707 + 0.707i)16-s + (−2.79 − 0.555i)18-s + (0.707 + 0.707i)19-s + (−0.923 + 0.382i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7119192569\)
\(L(\frac12)\) \(\approx\) \(0.7119192569\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.195 - 0.980i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
13 \( 1 + (-1.17 - 1.17i)T + iT^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.275 + 0.275i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
67 \( 1 + (-0.785 - 0.785i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 1.66T + T^{2} \)
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

−9.417482131684073477569483840676, −8.539841001918001526974162247694, −7.927888283968342764145413753267, −6.84166778660510457779676963622, −6.41558980335162284389234556991, −5.74443686125037583899267073696, −5.17581705596652995301242303538, −4.08449859349570615917540018194, −1.75446954098929112750711308888, −1.01818786768436410565491404853, 1.13544775185211412368775257157, 2.96398884726267826035963654230, 3.70109148455252281112921292852, 4.63321075548402588870009607755, 5.48925415890466596535621308166, 6.08577747489134654234015033965, 7.12885404528320080250841384646, 8.579856455121930263349840914986, 9.495210091199874261481811935203, 9.812609859896217708980456774838

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