↑

Properties

Label 2-2280-19.7-c1-0-29
Degree $2$
Conductor $2280$
Sign $0.999 - 0.00757i$
Analytic cond. $18.2058$
Root an. cond. $4.26683$
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  + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + 3.84·7-s + (−0.499 + 0.866i)9-s + 3.95·11-s + (1.74 − 3.02i)13-s + (0.499 − 0.866i)15-s + (−0.871 − 1.51i)17-s + (2.95 + 3.20i)19-s + (1.92 + 3.32i)21-s + (−0.570 + 0.988i)23-s + (−0.499 + 0.866i)25-s − 0.999·27-s + (2.15 − 3.73i)29-s + 0.0599·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s + 1.45·7-s + (−0.166 + 0.288i)9-s + 1.19·11-s + (0.483 − 0.837i)13-s + (0.129 − 0.223i)15-s + (−0.211 − 0.366i)17-s + (0.677 + 0.735i)19-s + (0.419 + 0.726i)21-s + (−0.118 + 0.206i)23-s + (−0.0999 + 0.173i)25-s − 0.192·27-s + (0.400 − 0.694i)29-s + 0.0107·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.999 - 0.00757i$
Analytic conductor: \(18.2058\)
Root analytic conductor: \(4.26683\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :1/2),\ 0.999 - 0.00757i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.526820522\)
\(L(\frac12)\) \(\approx\) \(2.526820522\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-2.95 - 3.20i)T \)
good7 \( 1 - 3.84T + 7T^{2} \)
11 \( 1 - 3.95T + 11T^{2} \)
13 \( 1 + (-1.74 + 3.02i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.871 + 1.51i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.570 - 0.988i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.15 + 3.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.0599T + 31T^{2} \)
37 \( 1 + 9.05T + 37T^{2} \)
41 \( 1 + (4.88 + 8.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.10 - 3.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.32 + 5.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.557 + 0.965i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.29 - 2.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.80 + 4.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.09 + 3.62i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.54 + 6.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.15 - 10.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.65 - 8.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.959T + 83T^{2} \)
89 \( 1 + (3.68 - 6.38i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.35 - 7.53i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.869790236405460170745616804386, −8.321133088263074344577200150555, −7.73446903777181140326268397796, −6.77519339705915149761816492014, −5.57695422440780381232926135424, −5.06642113101417304062870582080, −4.09022723978563222425364167685, −3.49576889904763115460974590446, −2.05221532524949253626131727493, −1.04646982657748523476922737302, 1.23507106240666238478772012524, 1.93878622625565916673968864529, 3.21845559551380485699432524506, 4.17652857362498982992814563027, 4.91022254869662169762550463707, 6.04421622654305795606548434383, 6.88137189954624145163899793601, 7.35159669490932829083737304457, 8.437514453842690252285964027032, 8.707568637390284260989357754940

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