↑

Properties

Label 1-2280-2280.563-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.385 - 0.922i$
Analytic cond. $10.5882$
Root an. cond. $10.5882$
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  − i·7-s − 11-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)17-s + (0.866 − 0.5i)23-s + (−0.5 − 0.866i)29-s + 31-s − i·37-s + (−0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + (−0.866 + 0.5i)47-s − 49-s + (−0.866 + 0.5i)53-s + (0.5 − 0.866i)59-s + (0.5 + 0.866i)61-s + ⋯
L(s)  = 1  − i·7-s − 11-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)17-s + (0.866 − 0.5i)23-s + (−0.5 − 0.866i)29-s + 31-s − i·37-s + (−0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + (−0.866 + 0.5i)47-s − 49-s + (−0.866 + 0.5i)53-s + (0.5 − 0.866i)59-s + (0.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.385 - 0.922i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.385 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9377027283 - 0.6244989173i\)
\(L(\frac12)\) \(\approx\) \(0.9377027283 - 0.6244989173i\)
\(L(1)\) \(\approx\) \(0.9539327634 - 0.04706049293i\)
\(L(1)\) \(\approx\) \(0.9539327634 - 0.04706049293i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 - iT \)
11 \( 1 - T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.85556948137524144054641497977, −19.03524577294425288918887097296, −18.399196549776484505295787265012, −17.567375477147828736418240251032, −16.98680206239925908892252540172, −16.163997357964174183487449891195, −15.56592634540247573045578256219, −14.77744810379621000955150258636, −13.83055566419534446827959210458, −13.281862596780049123581334578738, −12.8229204681478736908844692330, −11.54774487841032091936361506263, −11.0294056381690588222139539493, −10.34499837201526612134828332789, −9.61237566034242210416803308456, −8.54313082476765694881590793013, −8.07561201764401277951180827773, −6.95062978121852377395047830374, −6.6096379752114835846996802551, −5.411724677388051242238033525583, −4.69018315338638021950946275128, −3.80940246747544597215306811148, −3.07981172725468782403503497023, −1.91592184868259146479829263486, −1.03142312522805509906844752439, 0.40681658633157293548038028943, 1.80272001904485995784975410735, 2.65135120705787944023526707603, 3.29521416747184271451726774208, 4.55530713104777191221085392769, 5.20662962978117169743475628356, 6.02526879108765147871081396043, 6.714970082533744558740499171747, 7.84055791543647258750033065895, 8.407835982749320377113231443927, 9.14416208408665099287588791628, 9.95364531177223820387304697869, 10.88236071389996485035689275208, 11.40004743652714667405565312678, 12.30153492735586751655669867177, 13.12686788698538988313605019309, 13.4760763410206910832624012996, 14.62930203618119792306527522836, 15.3770518123184078843322518678, 15.751662789708627237749977663485, 16.51979834964482487425153396358, 17.616041991833748966717408131445, 18.09328910460019859889700037846, 18.76264827320336962691681189331, 19.34926577679871147045845009320

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