↑

Properties

Label 1-4800-4800.1331-r0-0-0
Degree $1$
Conductor $4800$
Sign $0.963 - 0.267i$
Analytic cond. $22.2911$
Root an. cond. $22.2911$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)7-s + (0.852 + 0.522i)11-s + (−0.522 − 0.852i)13-s + (0.951 − 0.309i)17-s + (−0.0784 + 0.996i)19-s + (−0.987 − 0.156i)23-s + (−0.760 − 0.649i)29-s + (0.309 + 0.951i)31-s + (0.972 + 0.233i)37-s + (0.156 + 0.987i)41-s + (−0.382 − 0.923i)43-s + (0.951 + 0.309i)47-s − i·49-s + (0.996 − 0.0784i)53-s + (−0.972 − 0.233i)59-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)7-s + (0.852 + 0.522i)11-s + (−0.522 − 0.852i)13-s + (0.951 − 0.309i)17-s + (−0.0784 + 0.996i)19-s + (−0.987 − 0.156i)23-s + (−0.760 − 0.649i)29-s + (0.309 + 0.951i)31-s + (0.972 + 0.233i)37-s + (0.156 + 0.987i)41-s + (−0.382 − 0.923i)43-s + (0.951 + 0.309i)47-s − i·49-s + (0.996 − 0.0784i)53-s + (−0.972 − 0.233i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $0.963 - 0.267i$
Analytic conductor: \(22.2911\)
Root analytic conductor: \(22.2911\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4800} (1331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4800,\ (0:\ ),\ 0.963 - 0.267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.091045570 - 0.2850434710i\)
\(L(\frac12)\) \(\approx\) \(2.091045570 - 0.2850434710i\)
\(L(1)\) \(\approx\) \(1.239027248 - 0.08856150831i\)
\(L(1)\) \(\approx\) \(1.239027248 - 0.08856150831i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (0.852 + 0.522i)T \)
13 \( 1 + (-0.522 - 0.852i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (-0.0784 + 0.996i)T \)
23 \( 1 + (-0.987 - 0.156i)T \)
29 \( 1 + (-0.760 - 0.649i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.972 + 0.233i)T \)
41 \( 1 + (0.156 + 0.987i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.996 - 0.0784i)T \)
59 \( 1 + (-0.972 - 0.233i)T \)
61 \( 1 + (0.233 + 0.972i)T \)
67 \( 1 + (-0.996 - 0.0784i)T \)
71 \( 1 + (0.453 + 0.891i)T \)
73 \( 1 + (0.987 + 0.156i)T \)
79 \( 1 + (0.951 + 0.309i)T \)
83 \( 1 + (0.0784 - 0.996i)T \)
89 \( 1 + (0.156 - 0.987i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.21385623032018422916642728175, −17.48197060273141325398223902430, −16.721823716509568868945721385702, −16.39666545801326118918151674103, −15.2104104586573911574223232568, −14.96311289614810251769215174185, −14.0541263885620463088880158090, −13.75450226524586295661940433813, −12.555731978699934517805779620855, −12.09882937490613747079909817053, −11.39563622540747046908411310594, −10.96692078959272976131010411320, −9.84109220021074202610109126015, −9.26121033640739604933270628990, −8.6968101045377167309887435575, −7.86505581798705164460461707006, −7.25095967904600370676619816634, −6.26335433931894310305699183758, −5.7647270009539933840846298432, −4.89673253711317769276373105159, −4.1864001598162929666720754632, −3.41778582348146507441440820377, −2.36922434235237835688806367459, −1.80582848814825364883664423658, −0.79156527806418023567894980866, 0.78013782821156861448563707878, 1.52807665199529975880681665698, 2.38024122530381213046429346788, 3.447142166670513958700809785599, 4.072697681932689536219861388275, 4.79499307857764126177735142568, 5.60645669662197427123601647938, 6.30887027861443413290164738456, 7.32393226281033812885217886958, 7.72331645071476773233416337399, 8.370736644337512075735140801434, 9.36686558716134061460442891468, 10.13988052901901044426300677050, 10.40228886731130345766032937798, 11.50303517087442849036132348108, 12.010221549334750578470711173966, 12.60117937282142869603918463395, 13.525854869795747561935119889916, 14.21646229505692301451339287845, 14.66970264930800430190614513556, 15.260810398547019412135291368998, 16.26596387478440632159744400212, 16.89301322486949364882058168458, 17.30697773144071599355179925534, 18.115577051583434512119322709518

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