Properties

Label 2-1520-19.11-c1-0-24
Degree $2$
Conductor $1520$
Sign $0.547 + 0.836i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
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  + (0.894 − 1.54i)3-s + (0.5 − 0.866i)5-s + 0.144·7-s + (−0.0989 − 0.171i)9-s + 5.35·11-s + (0.139 + 0.242i)13-s + (−0.894 − 1.54i)15-s + (−1.02 + 1.77i)17-s + (4.30 + 0.682i)19-s + (0.129 − 0.223i)21-s + (−1.98 − 3.44i)23-s + (−0.499 − 0.866i)25-s + 5.01·27-s + (1.69 + 2.94i)29-s + 4.97·31-s + ⋯
L(s)  = 1  + (0.516 − 0.894i)3-s + (0.223 − 0.387i)5-s + 0.0545·7-s + (−0.0329 − 0.0571i)9-s + 1.61·11-s + (0.0388 + 0.0672i)13-s + (−0.230 − 0.399i)15-s + (−0.248 + 0.429i)17-s + (0.987 + 0.156i)19-s + (0.0281 − 0.0488i)21-s + (−0.414 − 0.718i)23-s + (−0.0999 − 0.173i)25-s + 0.964·27-s + (0.315 + 0.546i)29-s + 0.894·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.547 + 0.836i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.547 + 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.444469530\)
\(L(\frac12)\) \(\approx\) \(2.444469530\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-4.30 - 0.682i)T \)
good3 \( 1 + (-0.894 + 1.54i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 0.144T + 7T^{2} \)
11 \( 1 - 5.35T + 11T^{2} \)
13 \( 1 + (-0.139 - 0.242i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.02 - 1.77i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.98 + 3.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.69 - 2.94i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.97T + 31T^{2} \)
37 \( 1 + 3.72T + 37T^{2} \)
41 \( 1 + (0.927 - 1.60i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.43 - 2.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.245 - 0.425i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.83 + 4.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.56 + 6.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.22 + 5.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.19 - 8.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.35 + 4.08i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.17 + 5.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.66 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + (-2.92 - 5.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.74 - 11.6i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.236335572994222876089897099417, −8.414339027907630574558356923091, −7.907101704595984497846658370590, −6.71969082001818199098112919408, −6.48899115552962724822888454894, −5.18276844955230411128101277011, −4.23136859617987607515088462925, −3.16384782959384238015569817843, −1.90118819029745969572108162732, −1.14921460066031805085775916728, 1.31380741600684764725411082491, 2.78477531359796432839210405784, 3.66311435835420917794965766917, 4.32651449728783781025388179014, 5.38388233176743433062316770880, 6.45256265743897919718379732435, 7.06346881896538749643305703881, 8.181438582043577841621021413402, 9.034962477470248660635723761695, 9.608505698958108577210564248840

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