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Properties

Label 1-2280-2280.1547-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

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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: $\chi_{2280} (1547, \cdot )$
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 \( 1 + (-0.866 + 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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