↑

Properties

Label 1-4800-4800.1811-r0-0-0
Degree $1$
Conductor $4800$
Sign $0.866 - 0.498i$
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 + (0.996 − 0.0784i)11-s + (0.0784 − 0.996i)13-s + (0.587 + 0.809i)17-s + (0.972 − 0.233i)19-s + (0.891 − 0.453i)23-s + (0.852 − 0.522i)29-s + (−0.809 + 0.587i)31-s + (0.649 + 0.760i)37-s + (0.453 − 0.891i)41-s + (0.382 + 0.923i)43-s + (0.587 − 0.809i)47-s − i·49-s + (−0.233 + 0.972i)53-s + (−0.649 − 0.760i)59-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)7-s + (0.996 − 0.0784i)11-s + (0.0784 − 0.996i)13-s + (0.587 + 0.809i)17-s + (0.972 − 0.233i)19-s + (0.891 − 0.453i)23-s + (0.852 − 0.522i)29-s + (−0.809 + 0.587i)31-s + (0.649 + 0.760i)37-s + (0.453 − 0.891i)41-s + (0.382 + 0.923i)43-s + (0.587 − 0.809i)47-s − i·49-s + (−0.233 + 0.972i)53-s + (−0.649 − 0.760i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $0.866 - 0.498i$
Analytic conductor: \(22.2911\)
Root analytic conductor: \(22.2911\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4800} (1811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4800,\ (0:\ ),\ 0.866 - 0.498i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.476832999 - 0.6619282343i\)
\(L(\frac12)\) \(\approx\) \(2.476832999 - 0.6619282343i\)
\(L(1)\) \(\approx\) \(1.385845583 - 0.1790915967i\)
\(L(1)\) \(\approx\) \(1.385845583 - 0.1790915967i\)

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 \( 1 + (0.996 - 0.0784i)T \)
13 \( 1 + (0.0784 - 0.996i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (0.972 - 0.233i)T \)
23 \( 1 + (0.891 - 0.453i)T \)
29 \( 1 + (0.852 - 0.522i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (0.649 + 0.760i)T \)
41 \( 1 + (0.453 - 0.891i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (-0.233 + 0.972i)T \)
59 \( 1 + (-0.649 - 0.760i)T \)
61 \( 1 + (0.760 + 0.649i)T \)
67 \( 1 + (0.233 + 0.972i)T \)
71 \( 1 + (0.987 - 0.156i)T \)
73 \( 1 + (-0.891 + 0.453i)T \)
79 \( 1 + (0.587 - 0.809i)T \)
83 \( 1 + (-0.972 + 0.233i)T \)
89 \( 1 + (0.453 + 0.891i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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

−18.29854332624912430303699559489, −17.5317140835193066434932195997, −16.842686428571154303960912473429, −16.20091741093040441999844684453, −15.569097196373850664631429368069, −14.59860538164872056788612586674, −14.33722824945604272953821429223, −13.69392295244377910810286120086, −12.6391437046274072816379681970, −12.043644290658329454075733874604, −11.428383521459649071315280529267, −11.05422411855225382692012323019, −9.81416514975361418886507819927, −9.24446471738332927894445209765, −8.85550083408949676262679952351, −7.80803153385799034267658762989, −7.25691912987518538641772991430, −6.448957570826668738407670476992, −5.630129581903132924894000269270, −4.98996763929175643224327107477, −4.23565344518746303661522811429, −3.38778173178653723281272749881, −2.5403785233782052641612050983, −1.639591649499213951450985294606, −0.98774599179616974010071536187, 0.97097185192044637824018275706, 1.21250286457910676248625361202, 2.4810768527648412641852482698, 3.366485652680223089189375606448, 4.00447499667762323474152238580, 4.82763439414427024804848763643, 5.53040856079850577492257597481, 6.34365502015223924792995802477, 7.15094299926399274757805196979, 7.77770860521007303161123629449, 8.445379264475850640235657725894, 9.18412366015325385832736149000, 10.04977628065971147881375513938, 10.6343636495149528133189673562, 11.27229180546464231914313227129, 11.99281413196465749958604046776, 12.694258618754374514320625103820, 13.42970026004811470617099811619, 14.181479415054591813379702009773, 14.629229790686348742104411533480, 15.31355459453288306796395457668, 16.14028608735233945204478641155, 16.89852287194445797444224025656, 17.37794622319795561831308171587, 17.91177414206548555512009782849

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