↑

Properties

Label 1-4800-4800.2531-r0-0-0
Degree $1$
Conductor $4800$
Sign $0.492 - 0.870i$
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 + (0.522 − 0.852i)11-s + (0.852 − 0.522i)13-s + (0.951 − 0.309i)17-s + (−0.996 − 0.0784i)19-s + (0.987 + 0.156i)23-s + (0.649 − 0.760i)29-s + (0.309 + 0.951i)31-s + (0.233 − 0.972i)37-s + (−0.156 − 0.987i)41-s + (−0.923 + 0.382i)43-s + (0.951 + 0.309i)47-s − i·49-s + (−0.0784 − 0.996i)53-s + (−0.233 + 0.972i)59-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)7-s + (0.522 − 0.852i)11-s + (0.852 − 0.522i)13-s + (0.951 − 0.309i)17-s + (−0.996 − 0.0784i)19-s + (0.987 + 0.156i)23-s + (0.649 − 0.760i)29-s + (0.309 + 0.951i)31-s + (0.233 − 0.972i)37-s + (−0.156 − 0.987i)41-s + (−0.923 + 0.382i)43-s + (0.951 + 0.309i)47-s − i·49-s + (−0.0784 − 0.996i)53-s + (−0.233 + 0.972i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $0.492 - 0.870i$
Analytic conductor: \(22.2911\)
Root analytic conductor: \(22.2911\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4800} (2531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4800,\ (0:\ ),\ 0.492 - 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.410335464 - 0.8228957184i\)
\(L(\frac12)\) \(\approx\) \(1.410335464 - 0.8228957184i\)
\(L(1)\) \(\approx\) \(1.073418631 - 0.1135082449i\)
\(L(1)\) \(\approx\) \(1.073418631 - 0.1135082449i\)

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 \( 1 + (0.522 - 0.852i)T \)
13 \( 1 + (0.852 - 0.522i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (-0.996 - 0.0784i)T \)
23 \( 1 + (0.987 + 0.156i)T \)
29 \( 1 + (0.649 - 0.760i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.233 - 0.972i)T \)
41 \( 1 + (-0.156 - 0.987i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (-0.0784 - 0.996i)T \)
59 \( 1 + (-0.233 + 0.972i)T \)
61 \( 1 + (-0.972 + 0.233i)T \)
67 \( 1 + (0.0784 - 0.996i)T \)
71 \( 1 + (-0.453 - 0.891i)T \)
73 \( 1 + (-0.987 - 0.156i)T \)
79 \( 1 + (0.951 + 0.309i)T \)
83 \( 1 + (0.996 + 0.0784i)T \)
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

−18.42385070508853685255761137796, −17.291221598832978712734579359102, −16.95519601934737451602598697983, −16.41179707027326207880845172429, −15.53647463336068726225930361283, −14.893320868611084299833110701917, −14.26455873252227441199338040480, −13.44975539724879543080522613225, −12.94843796865300120192319140831, −12.234043065118965774079336368262, −11.53651384462186121786072946882, −10.64850374733755281908281399625, −10.16781681121083705787033423797, −9.43397472766969833954929394382, −8.75249920587890787261949929530, −7.953276651076971250254433683179, −7.10427041843310322078372921410, −6.538802796726390743498492763742, −6.00134555393097687206933450468, −4.8266377819742613759913991930, −4.22383019457550268457484018610, −3.52448137475696127595228549080, −2.75807257662303428471734807556, −1.612491854042639172556761087780, −0.99501304275150848764261496587, 0.515562383104162368407678845763, 1.39842681807998896533731249689, 2.5315208027023272952786804881, 3.22671685048130803040728749749, 3.75975208847065152320901017210, 4.8469855791791022813363337568, 5.69690796057688869864953208203, 6.171043974382496781761256350868, 6.842548821805026467112799595817, 7.820478493401758864162935917038, 8.652762961225875465717748440868, 8.972336481396260240268607791917, 9.84796639751192445280070467933, 10.64105651957377905744522758325, 11.18165127049961287484507370147, 12.17126123285409199905284811005, 12.45431855317023905044971262505, 13.48377891934975264167431299756, 13.78518646979099380352378587190, 14.8741000946797843133000604466, 15.266785493064549812752601418370, 16.18885276337285979911576256619, 16.47076116466010431525483967924, 17.3637175672410379353972833569, 18.05392430213152209466468741766

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