Properties

Label 1-4560-4560.1307-r0-0-0
Degree $1$
Conductor $4560$
Sign $-0.303 - 0.952i$
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 + (−0.939 − 0.342i)13-s + (0.642 − 0.766i)17-s + (−0.984 + 0.173i)23-s + (0.642 + 0.766i)29-s + (−0.5 − 0.866i)31-s + 37-s + (0.939 − 0.342i)41-s + (−0.173 + 0.984i)43-s + (0.642 + 0.766i)47-s + (0.5 − 0.866i)49-s + (0.173 + 0.984i)53-s + (−0.642 + 0.766i)59-s + (0.984 − 0.173i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)7-s + (−0.866 − 0.5i)11-s + (−0.939 − 0.342i)13-s + (0.642 − 0.766i)17-s + (−0.984 + 0.173i)23-s + (0.642 + 0.766i)29-s + (−0.5 − 0.866i)31-s + 37-s + (0.939 − 0.342i)41-s + (−0.173 + 0.984i)43-s + (0.642 + 0.766i)47-s + (0.5 − 0.866i)49-s + (0.173 + 0.984i)53-s + (−0.642 + 0.766i)59-s + (0.984 − 0.173i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.303 - 0.952i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (1307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ -0.303 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8000030751 - 1.094695107i\)
\(L(\frac12)\) \(\approx\) \(0.8000030751 - 1.094695107i\)
\(L(1)\) \(\approx\) \(1.006298959 - 0.2303678012i\)
\(L(1)\) \(\approx\) \(1.006298959 - 0.2303678012i\)

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

−18.45343709934698260354520018718, −17.64928677541086130211055630499, −17.31747364545228926355403922169, −16.35608765229824304824813162851, −15.73465510746152882775206383016, −14.94482101111475485439849032507, −14.52317954175094324520226419645, −13.85899385435893553131092976135, −12.86404403473748982179441638853, −12.32640922986155412750785616426, −11.75938623277987658163518324240, −10.96184891195366705756398080320, −10.151499052243609732148843583554, −9.7284978329295733760137201452, −8.681559619804771246088564284508, −8.07086996846613346779700039165, −7.54928247037668941341884301987, −6.690254346055229708322256188496, −5.71784845662983100615056269296, −5.18818993249353549130985436703, −4.459152691152697743218714440238, −3.69007638465961516471783273954, −2.34741720731760504553592128531, −2.25695911346429668405701819474, −1.05999827602006750423950645020, 0.39449037684667520799181709497, 1.31129428802890248489585647006, 2.39335250034862377906199496573, 2.95093491314951610254033672541, 4.01846564388464044132104628372, 4.73893268310983687146493494883, 5.39015001353443853960342442375, 6.05807351219566377227849171081, 7.18865309264712951055495814204, 7.801798480751871052091520788202, 8.07121033698128411812056279696, 9.23418117871667971856102672127, 9.83689883388273908845772843866, 10.66388911145342413227578529669, 11.09143344375489324026449485230, 11.99078524852808285926444771382, 12.527855588332992876679612214357, 13.44697276536632831662769080984, 14.01240022997126283668453578975, 14.6090124942449407926464035610, 15.26974543469617403207508887953, 16.18010483257668678096677135591, 16.588801410264655402855761063586, 17.44393235566153170244844875661, 18.06456042860431443727394685988

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