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Properties

Label 1-2880-2880.59-r0-0-0
Degree $1$
Conductor $2880$
Sign $0.617 - 0.786i$
Analytic cond. $13.3746$
Root an. cond. $13.3746$
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  + (0.965 − 0.258i)7-s + (−0.608 − 0.793i)11-s + (0.793 + 0.608i)13-s − i·17-s + (0.923 − 0.382i)19-s + (0.965 + 0.258i)23-s + (−0.991 − 0.130i)29-s + (−0.5 − 0.866i)31-s + (0.923 + 0.382i)37-s + (0.965 + 0.258i)41-s + (0.608 + 0.793i)43-s + (−0.866 − 0.5i)47-s + (0.866 − 0.5i)49-s + (−0.382 + 0.923i)53-s + (−0.130 − 0.991i)59-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)7-s + (−0.608 − 0.793i)11-s + (0.793 + 0.608i)13-s − i·17-s + (0.923 − 0.382i)19-s + (0.965 + 0.258i)23-s + (−0.991 − 0.130i)29-s + (−0.5 − 0.866i)31-s + (0.923 + 0.382i)37-s + (0.965 + 0.258i)41-s + (0.608 + 0.793i)43-s + (−0.866 − 0.5i)47-s + (0.866 − 0.5i)49-s + (−0.382 + 0.923i)53-s + (−0.130 − 0.991i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(13.3746\)
Root analytic conductor: \(13.3746\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2880,\ (0:\ ),\ 0.617 - 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.785844219 - 0.8686042425i\)
\(L(\frac12)\) \(\approx\) \(1.785844219 - 0.8686042425i\)
\(L(1)\) \(\approx\) \(1.233311696 - 0.1930850520i\)
\(L(1)\) \(\approx\) \(1.233311696 - 0.1930850520i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.965 - 0.258i)T \)
11 \( 1 + (-0.608 - 0.793i)T \)
13 \( 1 + (0.793 + 0.608i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (0.965 + 0.258i)T \)
29 \( 1 + (-0.991 - 0.130i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (0.965 + 0.258i)T \)
43 \( 1 + (0.608 + 0.793i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (-0.130 - 0.991i)T \)
61 \( 1 + (-0.991 - 0.130i)T \)
67 \( 1 + (0.608 - 0.793i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (-0.130 + 0.991i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−19.18249781548564413340716745655, −18.36635671113107204125687028122, −17.913581581409074425309270729646, −17.34467257025795259593296818808, −16.39478368874687461502261941195, −15.69361463840472207367333209459, −14.91959456676026055730500023402, −14.55794967800940820933789840921, −13.5626714306434916543754250645, −12.77951096182623230613785643202, −12.33690049821658158093424589797, −11.1816538662053149655498353192, −10.88620138324110215118970553392, −10.0222995545848317925460582915, −9.12238969683643592441933255104, −8.38607947934245632123532795099, −7.70695350177808878198511920605, −7.10166876662452229946561432193, −5.871787040398293446902752761910, −5.41395752696230185617892054497, −4.56304316050795490977069345084, −3.713815797349930360638518116817, −2.7739141522638497069176902984, −1.820954563437766557635232169049, −1.0950134328521020663582306162, 0.701406890817890698043835101086, 1.531069890446863206137947658652, 2.60813734925488348945891228120, 3.39166836671713489411496364852, 4.36196269588446622008434181324, 5.099360225072181668102855603820, 5.775740119274035351934418818227, 6.71962220239147054949440342390, 7.69434034658909650427156257724, 7.98314752869472715113986430673, 9.1929230852811882317043764860, 9.44457089461562703491239811597, 10.849907167841713759765496580073, 11.17311009965705061598923278801, 11.64885712242892941790048219683, 12.820058777368385078591720351720, 13.58891763238008583859595874183, 13.93745588333215255864361552800, 14.83922003009295794317528824086, 15.54008994419944625005444317472, 16.37395487215683517835418844049, 16.79164223209674946460620013878, 17.82495675868325092629408043401, 18.37248138788261585657840881291, 18.8188617358256555944526137410

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