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Properties

Label 1-4560-4560.1523-r0-0-0
Degree $1$
Conductor $4560$
Sign $-0.944 + 0.328i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.944 + 0.328i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (1523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ -0.944 + 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1485808368 + 0.8793924289i\)
\(L(\frac12)\) \(\approx\) \(0.1485808368 + 0.8793924289i\)
\(L(1)\) \(\approx\) \(0.9386788092 + 0.2527799823i\)
\(L(1)\) \(\approx\) \(0.9386788092 + 0.2527799823i\)

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

−17.896626677944832620887263079874, −17.41282896019280396247519574633, −16.476970504052457605311626871728, −16.03413485241802775430314032853, −15.12144245737709198229266889530, −14.69828525969641480511744450921, −13.77833342699477660533387171892, −13.19327645823530794624412608764, −12.78366715529531496407524435692, −11.56932166307236442027311286106, −11.06273947517251815601632390998, −10.64971820063102236053153972479, −9.82498364810379869739387618714, −8.77690996965026188751507092712, −8.35893181176268704987898007311, −7.60564247735716913569792684627, −6.96163798197354542165762901010, −5.943583468976997862638970803442, −5.39807573535385460988982058359, −4.52178687068369680557184591712, −3.90550200102280531803819333200, −2.91840364305823255601671164347, −2.17500861467238260326240959665, −1.19979579977534106995082744492, −0.23333354943739141606380176385, 1.42104559172700804101127783084, 1.923186865859461446862675003534, 2.80496475724705571744423415325, 3.761430391300959246659720035540, 4.62519294753213301674036147608, 5.15296331381781709282015264926, 5.98314559402012555858179297295, 6.741825299520664290162665861418, 7.64471378652863354495357867812, 8.1322912468848112931554185320, 9.02527125609909941061938581178, 9.44950711353311849053172347196, 10.5298911280788611799013037757, 11.17362285385070820351617347616, 11.54212517467820987408569657410, 12.514211067589538927954427901587, 13.17570720129003884136454934582, 13.72274910668062584445509227354, 14.67754637610620895734979868521, 15.10448293231207456287408031436, 15.83749961971158024454683912322, 16.38471911364366718656150789064, 17.44228718935896759760138138213, 17.81321805626543701397204318958, 18.482677385007527922545359148197

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