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Properties

Label 1-4560-4560.203-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.716 - 0.697i$
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  + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)11-s + (−0.766 + 0.642i)13-s + (0.984 + 0.173i)17-s + (−0.342 + 0.939i)23-s + (−0.984 + 0.173i)29-s + (−0.5 − 0.866i)31-s − 37-s + (−0.766 − 0.642i)41-s + (−0.939 + 0.342i)43-s + (0.984 − 0.173i)47-s + (0.5 − 0.866i)49-s + (0.939 + 0.342i)53-s + (0.984 + 0.173i)59-s + (−0.342 + 0.939i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)11-s + (−0.766 + 0.642i)13-s + (0.984 + 0.173i)17-s + (−0.342 + 0.939i)23-s + (−0.984 + 0.173i)29-s + (−0.5 − 0.866i)31-s − 37-s + (−0.766 − 0.642i)41-s + (−0.939 + 0.342i)43-s + (0.984 − 0.173i)47-s + (0.5 − 0.866i)49-s + (0.939 + 0.342i)53-s + (0.984 + 0.173i)59-s + (−0.342 + 0.939i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.716 - 0.697i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7044694149 - 0.2864483939i\)
\(L(\frac12)\) \(\approx\) \(0.7044694149 - 0.2864483939i\)
\(L(1)\) \(\approx\) \(0.7779467923 + 0.03658835339i\)
\(L(1)\) \(\approx\) \(0.7779467923 + 0.03658835339i\)

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

−18.39569160222479541390985969412, −17.61790485026729430319746539047, −16.85591620434088999284533136075, −16.40194131742750004956818176213, −15.62788158404178276161361216799, −15.00519907508786929461449974859, −14.28543465851232049862822461694, −13.544324538555497852907673559653, −12.74620393374217389378532686140, −12.48241787685857861111588992036, −11.61456178592724923611038604052, −10.49677390901991829179572473396, −10.20947102618331620705962691158, −9.647664660242955280571175812193, −8.66970157726681163367553236241, −7.88430311159493929612064076316, −7.21480477195227563329317543144, −6.706520535994812579458584904157, −5.574381611610112181400113558284, −5.18760003886556160218826043791, −4.18367371363862457767945145107, −3.359359001612948870841121976897, −2.73819651877227639264849810188, −1.840241400996713276636627950069, −0.62885037541603803486555674197, 0.30518853875851631802263603848, 1.69812811244957843818800074451, 2.424315068259361032202493812027, 3.31220670959500989787528939273, 3.82211131402820533217577279221, 5.035166428614605994812577031555, 5.5983815994817814883964577113, 6.18332073882305655519167292055, 7.27738834240052439929159027427, 7.58873248366755119882026896214, 8.699689664830123197559403716885, 9.19322594056677531263809635920, 10.07227529178224690613459616965, 10.404486623946555082378797181228, 11.62258677284861381014539680631, 11.93093813664277688064243051434, 12.828016216107002542470263798027, 13.34581365762793511239697120814, 14.05235296159922625020739212770, 14.92424052937064063788710229881, 15.434731683582777292397017461880, 16.22395579097797373795938786903, 16.70647608850875903109660391128, 17.34977123431103517787555342199, 18.48924565321973596794744646684

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