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Properties

Label 1-4560-4560.3149-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.759 - 0.650i$
Analytic cond. $21.1765$
Root an. cond. $21.1765$
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  + (−0.5 + 0.866i)7-s + (0.866 − 0.5i)11-s + (−0.342 − 0.939i)13-s + (0.766 − 0.642i)17-s + (−0.173 + 0.984i)23-s + (−0.642 + 0.766i)29-s + (0.5 − 0.866i)31-s − i·37-s + (0.939 + 0.342i)41-s + (−0.984 + 0.173i)43-s + (0.766 + 0.642i)47-s + (−0.5 − 0.866i)49-s + (0.984 + 0.173i)53-s + (−0.642 − 0.766i)59-s + (0.984 + 0.173i)61-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (0.866 − 0.5i)11-s + (−0.342 − 0.939i)13-s + (0.766 − 0.642i)17-s + (−0.173 + 0.984i)23-s + (−0.642 + 0.766i)29-s + (0.5 − 0.866i)31-s − i·37-s + (0.939 + 0.342i)41-s + (−0.984 + 0.173i)43-s + (0.766 + 0.642i)47-s + (−0.5 − 0.866i)49-s + (0.984 + 0.173i)53-s + (−0.642 − 0.766i)59-s + (0.984 + 0.173i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.759 - 0.650i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (3149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.759 - 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.478164637 - 0.5468071154i\)
\(L(\frac12)\) \(\approx\) \(1.478164637 - 0.5468071154i\)
\(L(1)\) \(\approx\) \(1.055484246 - 0.05036019101i\)
\(L(1)\) \(\approx\) \(1.055484246 - 0.05036019101i\)

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 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.642 + 0.766i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (0.984 + 0.173i)T \)
59 \( 1 + (-0.642 - 0.766i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (-0.642 + 0.766i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−18.38244468197396426583213829189, −17.41142961351756577871685519935, −16.86789590590765006793069310188, −16.56595551687698536725751539850, −15.66029941225925627131352239857, −14.76418675560804002086699596063, −14.34715405241007368903284044915, −13.62246933683892122117887660802, −12.94238890207650355681852419645, −12.064307263553604204155875128387, −11.76563919412318310748283732303, −10.62844671669539509115473260616, −10.15497234012495765371719714494, −9.49096894404822837561414766440, −8.77026596760490340160053846169, −7.92095133578381742000009092162, −7.08525243759438365456475862260, −6.64421517666482014470479079010, −5.9211997232725190988409853762, −4.83196202816066539551352058469, −4.12404259125680311810515593467, −3.659113936238876191142207360605, −2.59521817615084306312351029679, −1.68263632755570697139045108722, −0.870533592902111963179340993095, 0.531897667141803531279651610424, 1.536785877616237188588365470406, 2.572656961547137923031458253834, 3.21491346717146027270720348039, 3.88515845844467693582987740070, 4.99624887271440409650023462340, 5.72937450360149347746860014918, 6.0984323835169114071651755111, 7.18278697254831166882098352777, 7.7462386965685238418393678229, 8.646046076148102312542674363426, 9.32444459662770837615104079784, 9.77775181227797725781067831710, 10.66925647186036294449898703706, 11.563248678880402462098859884814, 11.97206880487678553284547595220, 12.75945211517959846620434627992, 13.33417353418037926675678543848, 14.23195140040117493850302059084, 14.79733359278754881966739632171, 15.47845438586234308819009064663, 16.16035250090960154536985948266, 16.72954795088552663916195454779, 17.534734279426299457068349178220, 18.16512080560537376393458544654

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